mm-3929 === Subject: Re: average size of the antichain > A. D. Korshunov, The number of monotone Boolean functions, Problemy > Kibernet. No. 38, (1981), 5-108, 272. MR0640855 (83h:06013) > related stuff? If yes, have you got an electronic copy that you could > send? Google fails. (Otherwise I have to go to a library...) {1,2,...,n}; the limiting distribution is normal in k: AD Korshunov, I. Shmulevich, On the distribution of the number of monotone boolean functions relative to the number of lower units, Discrete Math, Volume 257, Issue 2-3 (November 2002), 463-479. It says: The results obtained here confirm the conjecture in [17] that the number of monotone Boolean functions relative to the number of terms in the minimal DNF asymptotically follows a normal distribution, with the assumption of all monotone Boolean functions being equiprobable. It includes a graph of the number of antichains with k elements, for n=7, with the average size ending up as 15. The mean for general n seems to be asymptotic to z_0 = 1/2 C(n, n/2) [1 - (n/2 - 1) / 2^(n/2)] for n even, and z_0 = 1/2 C(n, (n-1)/2) - C(n, (n-3)/2) (n - 1) 2^[-(n+1)/2-3] - C(n, (n-1)/2) (n - 3) 2^[-(n+1)/2 - 2], for n odd. coefficient. --- Christopher Heckman === Subject: Re: Mega M&Ms in a jar I (actually) WON!! There were 1,834 Mega M&M's in the jar so my guess of 1,821 was only 13 short. Grand Prize: 1,834 Mega M&M's !! === Subject: Re: Calculus Book differences are the increasing use of technology, > but most of these additions are found in selected > exercises, and those exercise may not be assigned > as HW anyway, so I wouldn't worry about that too > much. The new editions sometimes have better > diagrams as a result of the use of more > sophisticated computer graphing programs, > but as far as the main theory is concerned, > Calculus hasn't changed that much in the last > 30 years. More like 300 years, unless the instructor covers a fair amount of theory, in which case it's about 180 years. Dave L. Renfro === Subject: Re: Calculus Book > I am taking a calculus course at a local college. We have been told to > refer the book Calculus with Analytical Geometry by Larson 7th > edition. It's a very costly book. I am getting the 4th edition for a > very less price. Does anyone know how different 7th ed is from the 4th. Textbooks are a scam perpetrated on starving students. I can't imagine that the definition of the derivative has changed between editions. But by making a few minor changes, the author gets to make sure the students have to buy a brand new book instead of used. === Subject: Re: Calculus Book On Thu, 15 Sep 2005 19:39:13 -0700, fishfry >> I am taking a calculus course at a local college. We have been told to >> refer the book Calculus with Analytical Geometry by Larson 7th >> edition. It's a very costly book. I am getting the 4th edition for a >> very less price. Does anyone know how different 7th ed is from the 4th. >Textbooks are a scam perpetrated on starving students. I can't imagine >that the definition of the derivative has changed between editions. But >by making a few minor changes, the author gets to make sure the students >have to buy a brand new book instead of used. Yes, it is a scam, particularly given the prices for most new texts, but I think it's more the publishers rather than the authors who are the driving force for new editions. === Subject: Re: Calculus Book > On Thu, 15 Sep 2005 19:39:13 -0700, fishfry >I am taking a calculus course at a local college. We have been told to >refer the book Calculus with Analytical Geometry by Larson 7th >edition. It's a very costly book. I am getting the 4th edition for a >very less price. Does anyone know how different 7th ed is from the 4th. >>Textbooks are a scam perpetrated on starving students. I can't imagine >>that the definition of the derivative has changed between editions. But >>by making a few minor changes, the author gets to make sure the students >>have to buy a brand new book instead of used. > Yes, it is a scam, particularly given the prices for most new texts, > but I think it's more the publishers rather than the authors who are > the driving force for new editions. While I can't speak for all textbook authors, I can certainly attest to the fact that almost all the pressure to revise my texts comes from my publisher (who makes a heck of a lot more than I do off of every copy sold). of getting rich, IMO. Rick === Subject: fft or fourier transform I have a sensor that can measure a rectified waveform (s(t)) of a signal (v(t)). so, s(t) = abs(v(t)) the signal v(t) is a simple sinusoid: v(t) = c + a*sin(wt) where w is the frequency of the sinusoid, c is the offset, and a is the amplitude. My goal is to get the original signal v(t) from the measured signal s(t). My question is this: we collect the fft of the s(t), which is H(w). H(w) has a peak at w, 2*w, 3*w, but only the first 2 terms really matter. I need to know what is the relation between H(w), H(2*w) to v(t)? If I can get a relationship of H(w) = f(c, a), and H(2*w) = f(c,a), then I have 2 equations and 2 unknowns and I can sort of recover my original waveform, v(t). Actually, I don't even care about the offset c, I only need to recover the amplitude a. any help will be appreciated. === Subject: Re: Standard Deviation of PISA >The TRUTH is that in a year and a half of travelling all over that >territory, we never met ONE single Russian or Ukranian (and this >EXCLUDES jews, because jews are NOT descendants of the Rus as the >Russians and Ukranians--and Swedes--are), Actually, a large percentage of Russian Jews do have some mixed blood. >So scratch this statistic off your list. It simply AIN'T true. The Russians are lying to make their crime rate seem worse than it is? >Having >been mugged twice in Washington, DC, and NEVER in Moscow, I really >didn't feel so safe in our nation's capitol's despicable >nigger-infested subways a couple of weeks ago. Ever stop to think that with your attitude, you might be singled out as a primary target? I can probably come up with a Russian source for the numbers. Of course if you read Russian even less well than you read English, you won't be able to do much with the material. >There wasn't one >piece of graffiti ANYWHERE, the toilet seats hadn't been engraved by >latrinos and niggers advertising their gang affiliations, AND THERE >WERE MIRRORS IN THE BATHROOMS. >That's one thing I have yet to understand about muds--why is it that >every place niggers and latrinos alight, NONE of the bathroom mirrors >remain intact? There are no bathrooms in DC metro stations, nincompoop. (And no graffiti either). Oops. lojbab -- lojbab lojbab@lojban.org Bob LeChevalier, Founder, The Logical Language Group (Opinions are my own; I do not speak for the organization.) Artificial language Loglan/Lojban: http://www.lojban.org === Subject: Re: Any one has an idea about (xD)^n Operator === Subject: Re: A REPLY TO MR> ANDREW WILES PROOF OF FLT-step 1 george ghiata vi?t b?c th? c.97 n?i dung: > Theorem: > If X,Y,Z are pairwise relative prime natural numbers,n=any odd prime, > and if X,Y,Z are none divisible by n then the equation: > X^n+Y^n=Z^n is impossible. > The general PROOF: > STEP 1: > Statement1: Z>X and Z>Y > Proof: > If X^n+Y^n=Z^n then > Z^n>X^n from where we get Z>X > and Z^n>Y^n from where we get Z>Y > Statement2 : We can write X=B+Q > Y=B+P > X+Y=2*B+Q+P > Z=B+Q+P > where B,Q,P are natural numbers > Proof of the statement: > We take B=X+Y-Z > We write: Y-Z=-Q > X-Z=-P > Therefore: > B=X+(Y-Z) > B=X-Q > X=B+Q > B=Y+(X-Z) > B=Y-P > Y=B+P > B=X+Y-Z > B=(B+Q)+(B+P)-Z > B=2*B+Q+P)-Z > Z=B+Q+P > X+Y=(B+Q)+(B+P) > X+Y=2*B+Q+P hihi.very good . This is Felmat Y^x+B^x=Z^x with x>=2 === Subject: confusion on limit/integral does not exist, not defined, infinite? Hi all, Do the following three terms mean the same thing or not for limit and integrals? 1. Does not exist; 2. Not defined; 3. Infinite. Can you give some examples to help me clarify them? They are confusing to me. ---------------------------------------- Also, the series 1+1/2+1/3+1/4+ ... do you call it divergent? I feel it is perfectly convergent to a function log(n)+ some small constant 0.5772 ,... am I right? === Subject: Re: confusion on limit/integral does not exist, not defined, infinite? > Hi all, > Do the following three terms mean the same thing or not for limit and > integrals? > 1. Does not exist; > 2. Not defined; > 3. Infinite. No. First of all, limit and integral are defined. If we didn't have definitions for these terms, we couldn't talk about them. A limit might not exist. The standard example seems to be considering f(x) = sin (1/x), if x =/= 0, where the limit of f(x) as x approaches 0 does not exist. I guess an integral might not exist if the function is not defined on the interval, or if the limit of the Riemann sum does not exist. A limit can be infinite; for instance, the limit of 1/x^2 as x approaches 0. When an integral is infinite, it is usually said to diverge. > Also, the series 1+1/2+1/3+1/4+ ... > do you call it divergent? Yes. Some people say it diverges to infinity, which gives a little more information about how it behaves. > I feel it is perfectly convergent to a > function log(n)+ some small constant 0.5772 ,... am I right? Sort of. If you look at the limit of (1 + 1/2 + ... + 1/n) - ln n, as n goes to infinity, the limit is 0.5772..., which is Euler's Constant (sometimes called gamma or the Euler-Mascheroni constant; I almost certainly have botched the second name). --- Christopher Heckman === Subject: question about region of convergence for complex power series? Hi all, I am wondering if the division of two power series, differentiation and integration of power series change the region of convergence or not? Also, I have read the following from the book: The importance of the uniform convergence of a power series is that it retians uniform convergence property when either being differentiated and integrated insde its circle of convergence, thereby allowing it to be manipulated like an arbitrary function. This statement is strange to me: 1) Non-uniform convergence is not retained when being differentiated and integrated inside the region of convergence? Am I right? 2) Only uniform convergence allows for differentiation/integration inside the region of convergence? Non-uniform convergence does not allow such operations? 3) What is be manipulated like an arbitrary function? And why uniform convergence makes it so? Can you help me understand the book? === Subject: Re: question about region of convergence for complex power series? >I am wondering if the division of two power series, differentiation and >integration of power series change the region of convergence or not? Differentiation and integration do not change the radius of convergence, but they can affect convergence on the boundary. Multiplication can increase the radius, division can either increase or decrease it. Simple example: 1 and 1-x have infinite radius of convergence, series for 1/(1-x) (about 0) has radius 1. >Also, I have read the following from the book: >The importance of the uniform convergence of a power series is that it >retians uniform convergence property when either being differentiated >and integrated insde its circle of convergence, thereby allowing it to >be manipulated like an arbitrary function. >This statement is strange to me: It could probably be stated better. >1) Non-uniform convergence is not retained when being differentiated >and integrated inside the region of convergence? Am I right? Since power series always converge uniformly on compact sets inside the circle of convergence, let's talk more generally about convergence of sequences of analytic functions. All kinds of things can go wrong. For example, there is a sequence of polynomials that converges to 1 on the closed right half-plane and to 0 on the open left half-plane. Another sequence of polynomials f_n converges to 0 at every point, but (f_n)'(0) = 1. Neither could happen with uniform convergence on compact sets: f = lim f_n would have to be analytic, and (f_n)' would converge (uniformly on compact sets) to f'. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: question about region of convergence for complex power series? Hi all, I am wondering if the division of two power series, differentiation and integration of power series change the region of convergence or not? Also, I have read the following from the book: The importance of the uniform convergence of a power series is that it retians uniform convergence property when either being differentiated and integrated insde its circle of convergence, thereby allowing it to be manipulated like an arbitrary function. This statement is strange to me: 1) Non-uniform convergence is not retained when being differentiated and integrated inside the region of convergence? Am I right? 2) Only uniform convergence allows for differentiation/integration inside the region of convergence? Non-uniform convergence does not allow such operations? 3) What is be manipulated like an arbitrary function? And why uniform convergence makes it so? Can you help me understand the book? === Subject: Re: Revisiting my (alleged) proof of the Goldbach Conjecture > [...] > For example: Suppose the Goldbach were false and suppose it was false > with 8, the gist or crux of my proof would then focus on the > nonexistence of a Composite number that is 3 x 5 = 15. So if Goldbach > were false and false at 8 then 15 would not exist. > Not true. 15 would still exist, it would just factor differently, in a > way OTHER than the product of two prime numbers that add up to 8; maybe > it would be the product of THREE prime numbers, for instance. > --- Christopher Heckman Not true. The Goldbach Conjecture when a valid proof is given finds itself equivalent to elements inside the proof. Whenever a valid proof is given of a conjecture that elements inside the proof are equivalent to the statement of the proof. So what is the engine of the proof of Goldbach is the Chebyshev Theorem applied to special Composite numbers which have 2 and only 2 prime factors. So an equivalent statement of the Goldbach Conjecture which is every even number is the sum of two primes is equivalent to the altered Chebyshev Theorem that instead of between N and 2N exists a prime is the equivalent statement that between N and 2N exists a Composite number which has 2 and only 2 prime factors. So the equivalent statement to Goldbach Conjecture is the statement that between N and 2N exists a composite number which has 2 and only 2 prime factors. The equivalent statement to Twin Primes Conjecture is the Chebyshev Theorem that between N and 2N exists a prime. The equivalent statement to Goldbach Conjecture is between N and 2N exists a Composite which has 2 and only 2 prime factors. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Revisiting my (alleged) proof of the Goldbach Conjecture > [...] > For example: Suppose the Goldbach were false and suppose it was false > with 8, the gist or crux of my proof would then focus on the > nonexistence of a Composite number that is 3 x 5 = 15. So if Goldbach > were false and false at 8 then 15 would not exist. > Not true. 15 would still exist, it would just factor differently, in a > way OTHER than the product of two prime numbers that add up to 8; maybe > it would be the product of THREE prime numbers, for instance. > Not true. Not true. 15 is defined, for instance, in the Peano axioms, as S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(0))))))))))))))). This number is a natural number, by definition. It also has a prime factorization, since every natural number does. (This can be proven by induction.) These statements can be proven without mentioning even GC. > The Goldbach Conjecture when a valid proof is given finds > itself equivalent to elements inside the proof. Whenever a valid proof > is given of a conjecture that elements inside the proof are equivalent > to the statement of the proof. I have a hard time with the verbage, but I think you're saying that every statement inside of a valid proof is true. This is not necessarily true. For instance, when proving there are an infinite number of primes, if you use contradiction, you also have the statement There are a finite number of primes included in the proof, which means that there are an infinite number of primes, AND there are only a finite number of primes! You can only use _statements_ of theorems, lemmas, etc., from the actual proof, not anything from the proof itself. > So what is the engine of the proof of > Goldbach is the Chebyshev Theorem applied to special Composite numbers > which have 2 and only 2 prime factors. > So an equivalent statement of the Goldbach Conjecture which is every > even number is the sum of two primes is equivalent to the altered > Chebyshev Theorem that instead of between N and 2N exists a prime is > the equivalent statement that between N and 2N exists a Composite > number which has 2 and only 2 prime factors. This altered Chebyshev Theorem is actually true and trivial to prove. PROPOSITION: Between N and 2N exists a Composite number which has 2 and only 2 prime factors. Proof: Let p be a prime number between N/2 and N, which exists by Chebyshev's Theorem. Then N = 2(N/2) <= 2p <= 2N, so 2p is between N and 2N, and is a composite number which has exactly 2 prime factors, 2 and p. QED. > So the equivalent statement to Goldbach Conjecture is the statement > that between N and 2N exists a composite number which has 2 and only 2 > prime factors. Contrary to _The Hunting of the Snark_, what you say three times is not automatically true. > The equivalent statement to Twin Primes Conjecture is the Chebyshev > Theorem that between N and 2N exists a prime. Where is the proof of this? > The equivalent statement > to Goldbach Conjecture is between N and 2N exists a Composite which has > 2 and only 2 prime factors. Proof? (If this were true, the GC should be trivial to prove, because your post suggests that the equivalence is easy to prove, in which case I've done the hard work above.) --- Christopher Heckman === Subject: Re: Revisiting my (alleged) proof of the Goldbach Conjecture This altered Chebyshev Theorem is actually true and trivial to prove. PROPOSITION: Between N and 2N exists a Composite number which has 2 and only 2 prime factors. Proof: Let p be a prime number between N/2 and N, which exists by Chebyshev's Theorem. Then N = 2(N/2) <= 2p <= 2N, so 2p is between N and 2N, and is a composite number which has exactly 2 prime factors, 2 and p. QED. Yes that is a bit clever. So the proof of Goldbach falls something like this: Suppose false, then there exists at least one Array for which there is no prime pairing of a prime in the leftward column of the Array and a prime in the rightward column of the Array 8-Array 4 4 3 5 2 6 1 7 1 8 Which further implies that no Composite number which has 2 and only 2 prime factors exists for that specific Array. Contradiction end of proof of Goldbach. For the 8 Array the special Composite number is simply 15. So for Goldbach to be false implies the existence of an Array which has no prime pairing which implies the nonexistence of a Composite number which has 2 and only 2 prime factors and this is clearly false from the altered Chebyshev Theorem. Archimedes Plutonium www.iw.net/~a_pltuonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Revisiting my (alleged) proof of the Goldbach Conjecture (snip) > Not true. 15 is defined, for instance, in the Peano axioms, as > S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(0))))))))))))))). This number is a > natural number, by definition. It also has a prime factorization, since > every natural number does. (This can be proven by induction.) These > statements can be proven without mentioning even GC. > --- Christopher Heckman Stop being foolish. You are mixing philosophy with math. The Chebyshev Theorem says there *exists* a prime between N and 2N. I simply lift that concept of *exist* into a proof of Goldbach. If Chebyshev Theorem used a different concept instead of exist then I would lift that. The altered-Chebyshev Theorem is that there *exists* a Composite between N and 2N which has 2 and only 2 prime factors. For Goldbach to be false is tantamount to there being or existing an Array for which there is no special Composite between N and 2N. Get out of mathematics if you cannot wash yourself clean of philosophy. Get out of mathematics if you cannot see that you are in the wishy washy nit pick land of philosophy. Maybe there ought to be a flagging of people in sci.math and sci.logic who are wandering fools of philosophy and lost and in the wrong newsgroup. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Revisiting my (alleged) proof of the Goldbach Conjecture > This altered Chebyshev Theorem is actually true and trivial to prove. > PROPOSITION: Between N and 2N exists a Composite number which has 2 and > only 2 prime factors. > Proof: Let p be a prime number between N/2 and N, which exists by > Chebyshev's Theorem. Then > N = 2(N/2) <= 2p <= 2N, > so 2p is between N and 2N, and is a composite number which has exactly > 2 prime factors, 2 and p. QED. Isn't it possible that the only such composite 'between' N and 2N, as you define 'between' is 2N itself? Can it be proven that there's a composite prime between N and 2N which is distinct from either? That seems a more interesting result, if it could be shown. Ken === Subject: Re: a valid Goldbach proof Re: Revisiting my (alleged) proof of the Goldbach Conjecture I am confident that the Arrays can be so formed as to rule out duplicates and gaps so that the entire Natural Numbers can be uniquely represented in an Array formation. And whence that unique representation is begot, then the formal proof of Goldbach is a easy walk through the park. But with the passage of many hours I realized that it does not matter about duplicates. And I believe a proof should be streamlined. A proof that is short and concise is better than a proof that tacks on items not needed. In the Array formation it is important that there are no gaps and that every Natural Number is represented. But it is unimportant if some numbers are duplicated because in the proof of Goldbach it is suppose that an array does not have a prime pair and it is discharged as a contradiction by the Chebyshev Theorem that a special Composite number which has 2 and only 2 prime factors does not exist. So in formalizing the Arrays it matters not whether there are duplicates of Natural Numbers but only matters if there are gaps of Natural Numbers missing in the Arrays. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Baseball field > Darin, you could no doubt use the formula for an ellipse, or even a > super-ellipse, but if the problem were left up to me, I think I'd go to > the park, step off the distance as I walked along the expected > perimeter, and by knowing or getting the length of my stride, do it > that way. > Better but a little tedious: do it with string, or a chalk > line (which is just a roll of string of a measured length > coated with colored chalk). Stake out the perimeter, > run the string along the outside of the stakes, measure > the length. If you do it with a 100' chalk line, you > can just count how many times you need the line, then measure > the last bit manually. > There are also 100' tape measures, but that might be too > much to invest in for a one-time use. You'll probably > need the chalk line to lay out the fence anyway. A quicker method is to use a road wheel (don't know what the correct name is, I've always heard it called a road wheel). This is simply a small bicycle-wheel-like mounted on a shaft so that you can walk and push the wheel ahead of you. It has a circumference of a specific length (typically 3ft or 1m) and has a click-counter that is advanced by a stub mounted on one of the spokes. Zero the counter and walk the path of the fence you want to put up. At the end, read the counter, multiply by the circumference, and add the remainder (which is usually written along the edge of the wheel). You can rent them for dirt cheap. An alternative is to take any bicycle and put a mark on the tire and start with the mark either at the top or at the bottom. Walk the bike along the path and keep track of the number full rotations. Use a tape measure (a cloth one works best, but a metal one will work as well if it isn't too stiff) and measure the circumference of the wheel and the residual amount of circumference remaining after the last full rotation (or just round up to the next full rotation). That will be more than accurate for your needs and is trivially easy to do. === Subject: Re: Baseball field > Darin, you could no doubt use the formula for an ellipse, or even a > super-ellipse, but if the problem were left up to me, I think I'd go to > the park, step off the distance as I walked along the expected > perimeter, and by knowing or getting the length of my stride, do it > that way. Better but a little tedious: do it with string, or a chalk line (which is just a roll of string of a measured length coated with colored chalk). Stake out the perimeter, run the string along the outside of the stakes, measure the length. If you do it with a 100' chalk line, you can just count how many times you need the line, then measure the last bit manually. There are also 100' tape measures, but that might be too much to invest in for a one-time use. You'll probably need the chalk line to lay out the fence anyway. | A quicker method is to use a road wheel (don't know what the | correct name is, I've always heard it called a road wheel). This | is simply a small bicycle-wheel-like mounted on a shaft so that | you can walk and push the wheel ahead of you. It has a | circumference of a specific length (typically 3ft or 1m) and has | a click-counter that is advanced by a stub mounted on one of the | spokes. Zero the counter and walk the path of the fence you want | to put up. At the end, read the counter, multiply by the | circumference, and add the remainder (which is usually written | along the edge of the wheel). You can rent them for dirt cheap. | An alternative is to take any bicycle and put a mark on the tire | and start with the mark either at the top or at the bottom. Walk | the bike along the path and keep track of the number full | rotations. Use a tape measure (a cloth one works best, but a | metal one will work as well if it isn't too stiff) and measure | the circumference of the wheel and the residual amount of | circumference remaining after the last full rotation (or just | round up to the next full rotation). That will be more than | accurate for your needs and is trivially easy to do. Anybody who reads this newsgroup and measures the circumference of a wheel with a tape instead of calculating it (pi and all that) should be hunted down and shot like dawgs !! _____Gerard S. === Subject: Re: Baseball field > Darin, you could no doubt use the formula for an ellipse, or even a > super-ellipse, but if the problem were left up to me, I think I'd go to > the park, step off the distance as I walked along the expected > perimeter, and by knowing or getting the length of my stride, do it > that way. > Better but a little tedious: do it with string, or a chalk > line (which is just a roll of string of a measured length > coated with colored chalk). Stake out the perimeter, > run the string along the outside of the stakes, measure > the length. If you do it with a 100' chalk line, you > can just count how many times you need the line, then measure > the last bit manually. There are also 100' tape measures, but that might be too > much to invest in for a one-time use. You'll probably > need the chalk line to lay out the fence anyway. > | A quicker method is to use a road wheel (don't know what the > | correct name is, I've always heard it called a road wheel). This > | is simply a small bicycle-wheel-like mounted on a shaft so that > | you can walk and push the wheel ahead of you. It has a > | circumference of a specific length (typically 3ft or 1m) and has > | a click-counter that is advanced by a stub mounted on one of the > | spokes. Zero the counter and walk the path of the fence you want > | to put up. At the end, read the counter, multiply by the > | circumference, and add the remainder (which is usually written > | along the edge of the wheel). You can rent them for dirt cheap. > | An alternative is to take any bicycle and put a mark on the tire > | and start with the mark either at the top or at the bottom. Walk > | the bike along the path and keep track of the number full > | rotations. Use a tape measure (a cloth one works best, but a > | metal one will work as well if it isn't too stiff) and measure > | the circumference of the wheel and the residual amount of > | circumference remaining after the last full rotation (or just > | round up to the next full rotation). That will be more than > | accurate for your needs and is trivially easy to do. > Anybody who reads this newsgroup and measures the circumference > of a wheel with a tape instead of calculating it (pi and all > that) should be hunted down and shot like dawgs !! _____Gerard S. Dawgs that would have produced a result with less than one-third and probably more like one-tenth the error. === Subject: question about number of roots in a certain region? Hi all, I have read that the number of complex roots of polynomial P(z) in a closed contour can be determined by N=1/(2*pi*i) * ContourIntegrateAlongTheContour(P'(z)/P(z)), where P'(z) is the derivitive of P(z). I want to find the number of roots of P(z)=z^5+3*z+18 in the first quadrant of z-plane. I evaluate the following: N=1/(2*pi*i) * ContourIntegrateAlongTheContour((5*z^4+3)/(z^5+3*z+18)), Here are the roots of P(z): >> roots([1 0 0 0 3 18]) ans = 1.4765 + 1.1428i 1.4765 - 1.1428i -0.6415 + 1.6372i -0.6415 - 1.6372i -1.6700 The ContourIntegrateAlongTheContour((5*z^4+3)/(z^5+3*z+18)) = (5* (1.4765 + 1.1428i)^4 + 3) = -50.1199 +29.5028i So it does not give an integer result. What's wrong? btw, I found this method to be suspious: in order to evaluate the contour integral, I have to find the roots and do the partial fraction expansion, but if I can find the roots, I already know the number of the roots in any region on the complex plane... so I don't need to know the number of roots in any contour any more. It is already known... So what's wrong with the method? -=------------------------------------- Also I'd like to ask: I can use Rouche's thoerem to determine the number of zeros in a circular region... If we restrict the roots to be real, how do I know the number of zeros/roots in an interval on the real axis? === Subject: Re: question about number of roots in a certain region? > Hi all, > I have read that the number of complex roots of polynomial P(z) in a > closed contour can be determined by > N=1/(2*pi*i) * ContourIntegrateAlongTheContour(P'(z)/P(z)), > where P'(z) is the derivitive of P(z). > I want to find the number of roots of P(z)=z^5+3*z+18 in the first > quadrant of z-plane. > I evaluate the following: > N=1/(2*pi*i) * ContourIntegrateAlongTheContour((5*z^4+3)/(z^5+3*z+18)), > Here are the roots of P(z): >> roots([1 0 0 0 3 18]) > ans = > 1.4765 + 1.1428i > 1.4765 - 1.1428i > -0.6415 + 1.6372i > -0.6415 - 1.6372i > -1.6700 > The ContourIntegrateAlongTheContour((5*z^4+3)/(z^5+3*z+18)) > = (5* (1.4765 + 1.1428i)^4 + 3) > = -50.1199 +29.5028i > So it does not give an integer result. What's wrong? > btw, I found this method to be suspious: in order to evaluate the > contour integral, I have to find the roots and do the partial fraction > expansion, but if I can find the roots, I already know the number of > the roots in any region on the complex plane... so I don't need to know > the number of roots in any contour any more. It is already known... > So what's wrong with the method? > -=------------------------------------- > Also I'd like to ask: I can use Rouche's thoerem to determine the > number of zeros in a circular region... > If we restrict the roots to be real, how do I know the number of > zeros/roots in an interval on the real axis? Missing information: what contour? And was it closed? And was it parametrized properly? And was the differential of the parametrization included? My calculations: First, find an upper bound m for the magnitude of the roots. (Any norm of the companion matrix; I found m=18 - and this a huge overestimate.) Then parametrize the boundary path of the square with vertices (0,0), (m,0), (m,m), (0,m) in this order. The Trapezoidal Rule for the contour integral N, with 100 subintervals of each edge gives (MATLAB used): N is approximated by 1 + (-6.631090327391398e-006 -2.387644718504380e-005i) which is reasonably close to 1. No calculation of the actual roots was needed. Remark: The questions about the missing information were prompted by my first few attempts to program the contour integral - I kept forgetting some details. === Subject: Re: question about number of roots in a certain region? >I have read that the number of complex roots of polynomial P(z) in a >closed contour can be determined by >N=1/(2*pi*i) * ContourIntegrateAlongTheContour(P'(z)/P(z)), >where P'(z) is the derivitive of P(z). Correct. >I want to find the number of roots of P(z)=z^5+3*z+18 in the first >quadrant of z-plane. >I evaluate the following: >N=1/(2*pi*i) * ContourIntegrateAlongTheContour((5*z^4+3)/(z^5+3*z+18)), On which contour? >Here are the roots of P(z): > roots([1 0 0 0 3 18]) >ans = > 1.4765 + 1.1428i > 1.4765 - 1.1428i > -0.6415 + 1.6372i > -0.6415 - 1.6372i > -1.6700 >The ContourIntegrateAlongTheContour((5*z^4+3)/(z^5+3*z+18)) >= (5* (1.4765 + 1.1428i)^4 + 3) >= -50.1199 +29.5028i No. Note that if f(z) = g(z)/h(z) has a simple pole at z=a, with g and h analytic there, g(a) <> 0 and h(a) = 0, then Res(f; a) = g(a)/h'(a). In your case g = P' and h = P, so the residue would be 1. >btw, I found this method to be suspious: in order to evaluate the >contour integral, I have to find the roots and do the partial fraction >expansion, but if I can find the roots, I already know the number of >the roots in any region on the complex plane... so I don't need to know >the number of roots in any contour any more. It is already known... No. You can e.g. evaluate a contour integral by numerical methods, without any need to find roots. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Hello everybody, **** System Error Number 175 I've tried looking everywhere, but I still can't find what it means.... I should mention that my code works perfectly in windows (I've tried it at school). So, does anybody know what this error means, or where I can find information about it?? The details of the problem I'm solving aren't that important, I'm just trying to find a the list of system errors and their meaning. rodrigo === I don't exactly know where this number is coming from. Syntax errors are listed in gamserrs.txt. OS errors are listed in /usr/include/asm/errno.h. Note that there are two 32 bit Linux versions: > With 32-bit Linux (Intel Linux) you can choose between two > different versions: > lxigams_sfx.exe for version 2.2 of the GNU C Library > lx3gams_sfx.exe for version 2.3 of the GNU C Library > The newer version, lx3gams_sfx.exe, is recommended. There is also a 64-bit Linux version which obviously does not run on a 32-bit OS. If the problem persists, send the details of your OS and the exact message to support@gams.com. If this message appears in the listing file, attach the listing file. The more information you provide, the better the problem can be diagnosed. > Hello everybody, > **** System Error Number 175 > I've tried looking everywhere, but I still can't find what it means.... > I should mention that my code works perfectly in windows (I've tried it > at school). So, does anybody know what this error means, or where I can > find information about it?? > The details of the problem I'm solving aren't that important, I'm just > trying to find a the list of system errors and their meaning. > rodrigo ---------------------------------------------------------------- Erwin Kalvelagen erwin@gams.com, http://www.gams.com/~erwin ---------------------------------------------------------------- === Subject: Re: Prime cf's and related fractions! Am 16.09.05 00:17 schrieb Proginoskes: > Let alpha = [a_0, a_1, a_2, ...], and p_k and q_k the numerator and > denominator of [a_0, a_1, ..., a_k], reduced to lowest terms. > Define p_(-1) = 1, q_(-1) = 0, p_0 = a_0, and q_0 = 1. Then > p_k = a_k p_(k-1) + p_(k-2), and > q_k = a_k q_(k-1) + q_(k-2), when k >= 1. This can even been written more concise as a matrix-problem: set P=[[1,0],[0,1]]; or if you have a first coefficient cf(x)=[a:b,c,d...] of a then set P=[[1,a],[0,1]] then set for each step another matrix Q_i as Q_i = [[0,1],[1,A(i)]] where A(i) gives the i'th coefficient of the continued fraction Then compute R = P * Q_1 * Q_2 * Q_3 * .... to the desired precision for instance by recursive computation R = P R = R * Q_1 R = R * Q_2 R = R * Q_3 ... The result is then between R[1,1]/R[2,1] <= result <= R[1,2]/R[2,2] If the coefficients of the continued fraction are periodic, say with period-length of k then compute first S = Q_1 * Q_2 *...* Q_k and apply then R = P * S * S * S ... Assume, you had the correct result, then it would be true that R = R * S which is an eigenvalue-problem on S, and can be solved simply. For periodic cf's this even shows, that the solution is a root of a polynomial of degree 2,thus a square-root. >>Is there a closed form for finding a numerator >>that does not require the previous numerator? Surely not if the coefficents cannot be expressed as a sequence with a polynomial generation rule (and even then it should be difficult). Gottfried Helms === Subject: Re: Prime cf's and related fractions! Sorry; I should correctmy previos post. Am 16.09.05 08:22 schrieb Gottfried Helms: > If the coefficients of the continued fraction are periodic, > say with period-length of k then compute first > S = Q_1 * Q_2 *...* Q_k > and apply then > R = P * S * S * S ... > Assume, you had the correct result, then it would be true that > R = R * S d*R = R * S // add a factor d; the quotients of the rows are not affected > which is an eigenvalue-problem on S, and can be solved simply. > For periodic cf's this even shows, that the solution is a root of > a polynomial of degree 2,thus a square-root. Gottfried Helms === Subject: Re: Using the maximum likelihood estimation method > Hi everyone, i am new to this forum and i have a math problem regarding using Maximum likelihood estimation. > I have a model equation with several parameters that i wish to verify using the MLE method. > The model equation is as followed: > P = Po*{Pa1*(V^Kpv1)*[1 + Kpf1(f - fo)]+(1-Pa1)*(V^Kpv2) > the parameters i wish to verify is: > Po > Pa1 > Kpv1 > Kpf1 > fo > Kpv2 > P, V and f refers to data collected, they are power, voltage and freq and is already what i have. > I am studying one of the papers that use the method of MLE but i am unsure of the steps, so please kindly give me some advice. > The MLE function is usually solved by taking the log of the likelihood function,and then solve the log-likelihood functin by taking the partial diff of the individual parameter isn't it??The log likelihood function is as follwed: > R(k) = cov(e(k),e(k)) Don't understand why you need this. > e(k) = Ycalc(k) - Ymeas(k) or e= P - Pmeasered for each pair of values f, V This is the error and if your errors are normally (Gaussian) distributed then the likelihood function is - Product over k exp(-.5*e(k)^2/sigma^2) * 1/sqrt(2pi*sigma^2 pi) taking the log because it is easier to minimise and you have (sum over k of -e(k)^2) -2*N*log(sigma) (check my algebra) It is this that you minimise wrt the parameters Po etc etc and sigma If you are lucky you will be able to get explicit expressions for some of these, the rest will have to be done numerically. > where Ycalc is the calculated output and Ymeas is the measured output. > V(delta)=(1/2)*N *[e(k)'*e(k)] + (1/2)*m*N*log(2*pi) > This is the result from the likelihood equation after taking the log of the function. > Now the part that i don understand is how do i fit in my model equation this log-likelihood function??? Also,the > R(k) and e(k), what are this two function for and how do i relate it to my model equation. > Please kindly enlighten me. > God bless. === Subject: Re: Almost done <4329752B.C2B68034@lmcinvestments.comsnip > 61. What is the greatest integer that > divides p^4-1 for every prime number p>5 Sorry P^4-1 is not equal to p^3. By PEMDAS exponentiation takes > precedence over everything but parentheses, spaces are irrelevant. Ohsurespacesareirrelevant,neededonlyforvisualeasybutotherthanthatawasteoftim e. === Subject: Re: Does n*zeta(n) = this sum? >> Does, for every n = integer >= 3, >> sum{j=2 to n-1} sum{k=1 to inf} zeta(j,k) * zeta(n+1-j,k) Your off on your factor of n but I will show you the general method: go ahead and plug in your hurwitz equation into your double sum and you will see that you get the sum on j factored out(Because the j's cancel in the exponent's of the two factors) and so you end up with (n-2)*sum(sum(1/(m+k)^(n+1),k=1..oo),m=0..oo) this sum may look very complicated but notice how symmetrical it seems. to find the some you need to notice it is equivilent to sum(k*1/k^(n+1),k=1..infinity) = zeta(n) the trick is to write out the terms as if they were the elements of a matrix: i.e. A[i,j] = 1/(i+j)^(n+1) an notice that the normal sum is just adding up the rows where each row looks like a hurwitz function (and each column) but look at it a different way. along anti-diagonals and you will notice the elements are constant. and each one grows one extra element as you move downward along the main diagonal. hell, I might as well do it ;) I will only do it for terms A[i,j] = i+j to make it easy but its same principle 123 4 56789.. 23 4 56789... 3 4 56789.... 4 56789..... ...... ...... ...... (I did the spacing on the 4 so you would notice it) notice though it is a similar idea to cantor's diagonal proof and we we get something like 1 + 2*2 + 3*3 + 4*4 + 5*5 + ..... or sum(k*A[0,k]) Its the same principle with any A[i,j] that exhibits this symmetry and you are left with a simple sum that I pointed out above. (and for n = 2 you get 0, which is expected.. ) Jon === Subject: Re: Point Inside a triangle > What is a good formula to use to decide whether or not a point is > inside a triangle, or where can I find one that is relatively simple to > understand. This is my best shot: http://huizen.dto.tudelft.nl/deBruijn/programs/suna57.htm Implementation issues (: Fortran) somewhat outdated, maybe. Han de Bruijn === Subject: Re: Point Inside a triangle !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> What is a good formula to use to decide whether or not a point is >> inside a triangle, or where can I find one that is relatively simple to >> understand. > This is my best shot: > http://huizen.dto.tudelft.nl/deBruijn/programs/suna57.htm It's what we'd call shooting with cannons on birds. For a triangle ABC and a point P, just check that the determinants |A-P B-P|, |B-P C-P| and |C-P A-P| have the same nonzero sign. Then P lies in the triangle. No slopes, divisions, singularities, whatever, and the most complicated involved operation is a multiplication. You might want to check the FAQ of comp.graphics.algorithms for problems like that. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Point Inside a triangle >What is a good formula to use to decide whether or not a point is >inside a triangle, or where can I find one that is relatively simple to >understand. >>This is my best shot: >>http://huizen.dto.tudelft.nl/deBruijn/programs/suna57.htm > It's what we'd call shooting with cannons on birds. That is so, because _my_ Inside/Outside of a triangle served another purpose than just that (: Efficient Point Probes). But do _you_ know in what context the original poster is going to use the technique? Han de Bruijn === Subject: Re: Point Inside a triangle !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >>What is a good formula to use to decide whether or not a point is >>inside a triangle, or where can I find one that is relatively simple to >>understand. >This is my best shot: >http://huizen.dto.tudelft.nl/deBruijn/programs/suna57.htm >> It's what we'd call shooting with cannons on birds. > That is so, because _my_ Inside/Outside of a triangle served > another purpose than just that (: Efficient Point Probes). The mathematics of the case does not change because of the purpose you are employing it for. > But do _you_ know in what context the original poster is going to > use the technique? Let's just assume that he is not lying when he says that he is going to use it for deciding whether or not a point is inside a triangle, and let's further assume that he is not lying when he claims to be looking for one that is relatively simple to understand. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Point Inside a triangle >>This is my best shot: >>http://huizen.dto.tudelft.nl/deBruijn/programs/suna57.htm >It's what we'd call shooting with cannons on birds. >>That is so, because _my_ Inside/Outside of a triangle served >>another purpose than just that (: Efficient Point Probes). > The mathematics of the case does not change because of the purpose you > are employing it for. Read the _whole_ thing before you start blathering about anything. Hint: how would you define the distance from a point to a triangle? Han de Bruijn === Subject: Re: Point Inside a triangle !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >This is my best shot: http://huizen.dto.tudelft.nl/deBruijn/programs/suna57.htm >>It's what we'd call shooting with cannons on birds. >That is so, because _my_ Inside/Outside of a triangle served >another purpose than just that (: Efficient Point Probes). >> The mathematics of the case does not change because of the purpose >> you >> are employing it for. > Read the _whole_ thing before you start blathering about anything. Why should I read pages of arguments and Fortran code for a problem with a solution that can be given in a single sentence? > Hint: how would you define the distance from a point to a > triangle? Why would you bother if the task at hand is deciding whether a point is within the triangle, in a relatively simple way? You are obviously proud to have a hammer, but not every problem is a nail. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Point Inside a triangle >>Hint: how would you define the distance from a point to a >>triangle? > Why would you bother if the task at hand is deciding whether a point > is within the triangle, in a relatively simple way? Sure. Why bother? As usual with you .... A summary before I quit here: The distance from a point to a triangle can be defined if you have the shape functions A,B,C of the triangle and define the distance function as min(A,B,C). Maybe my implementation and the (: old) story around it is somewhat clumsy but this is all, essentially. Not difficult at all. Han de Bruijn === Subject: Re: Point Inside a triangle !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >Hint: how would you define the distance from a point to a >triangle? >> Why would you bother if the task at hand is deciding whether a point >> is within the triangle, in a relatively simple way? > Sure. Why bother? As usual with you .... A summary before I quit here: > The distance from a point to a triangle can be defined if you have > the shape functions A,B,C of the triangle and define the distance > function as min(A,B,C). Maybe my implementation and the (: old) > story around it is somewhat clumsy but this is all, essentially. Not > difficult at all. Except that shape function is not a basic mathematical concept, so your summary isn't one. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Point Inside a triangle >>What is a good formula to use to decide whether or not a point is >>inside a triangle, or where can I find one that is relatively simple to >>understand. >(Assumed: the vertices are given by Cartesian coordinates. If not, specify >otherwise.) >The triangle having vertices A,B,C (listed counterclockwise) and the point >being P, calculate the oriented areas of triangles >ABP, BCP, CAP >- if they are all positive, P is inside. Else, outside or on the boundary. >Using cross-product (embedding the triangle in the xy-plane of the >xyz-space): Oriented area of ABP is the z-coordinate of > (B-A) cross (P-A). >It is intuitive enough to figure out the conditions of P being on the >boundary. >>If the vertices are P1(x1,y1), P2(x2,y2), P3(x3,y3) numbering >>counterclockwise, you can check that the point P(x,y) lies to the left >>of the line P1-P2, and of P2-P3, and of P3-P1. >>To determine which side of the line the point falls on is simple. For P >>on the line P1-P2, for example, we have >>y-y1 = y2-y1 >>---- ----- >>x-x1 x2-x1 >>or, more correctly, (allowing for the case x = x1) >>S = (y-y1)(x2-x1) - (x-x1)(y2-y1) = 0 >>P lies on the left of P1-P2 when S > 0. Therefore you just need to >>evaluate S for the three sides, and if it is > 0 in each case the point >>is inside the triangle. (Of course, > 0 for two sides and = 0 for the >>third means the point falls on the third side, and = 0 for two sides >>puts P at a vertex.) > I don't see how this will work... knowing that P is to the left of > all the edges doesn't mean it's inside the triangle. In my 2D FE programs I always use the convention that element nodes are listed in counterclockwise order. Given this condition, the determination of insideness by checking left-side-ness of the point with respect to each line is the fastest method I've seen. === Subject: continuity terminology In probability, we have a cumulative distribution function (CDF) Fx(x)=|0 x<1 |0.3 x>=1 |0.6 x>=2 |1.0 x>=3 whose probability mass function (PMF) is Px(x)=|0.3 x=1 |0.3 x=2 |0.4 x=3 |0 otherwise The teacher says the PMF is a discrete function because it has values only at a finite set of points in its domain, but the CDF is continuous because it has values at every point in its domain. According to my calculus book, this CDF is not continuous, because, while there is a value for every point in the domain, the limit approaching that point doesn't exist because the limit approaching from below is not equal to the limit approaching from above for, say, x=1. Is this a special meaning of 'continuous', or is there another word, opposite to 'discrete', to describe the CDF? -- john === Subject: Re: continuity terminology 09/16/2005 >The teacher says the PMF is a discrete function because it has values >only at a finite set of points in its domain, but the CDF is >continuous because it has values at every point in its domain. Ask your teacher to define continuous. Whether the CDF is continuous depends on what definition she is using. >but the CDF is continuous >because it has values at every point in its domain. And if its domain had been {0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1}? -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: Joke on limits //I don't get this joke. Can somebody explain it to me if you http://www.sgoc.de/math.html //That's where I found it. 1 + 1 = 3, for large values of 1 http://www.sgoc.de/Pics/limes.gif //Picture of the proof. Limit of square root of 8 as x goes from 8 to 9 equals 3. limit of 3 as w goes to infinity equals 8. === Subject: Re: Joke on limits >//I don't get this joke. Can somebody explain it to me if you >http://www.sgoc.de/math.html //That's where I found it. >1 + 1 = 3, for large values of 1 >http://www.sgoc.de/Pics/limes.gif //Picture of the proof. Limit of >square root of 8 as x goes from 8 to 9 equals 3. No! It's the limit of the square root of 8 as _8_ tends to 9. >limit of 3 as w goes >to infinity equals 8. Here it's important that it's omega, not w. (What do you do to the symbol for omega to turn it into the symbol for infinity?) ************************ === Subject: Re: Joke on limits > //I don't get this joke. Can somebody explain it to me if you > http://www.sgoc.de/math.html //That's where I found it. > 1 + 1 = 3, for large values of 1 Wouldn't you say 1.5 is a large value of 1? > http://www.sgoc.de/Pics/limes.gif //Picture of the proof. Limit of > square root of 8 as x goes from 8 to 9 equals 3. limit of 3 as w goes > to infinity equals 8. It's important to write it correctly. It should read: limit_{w->oo} 3 = 8 -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Joke on limits My favorite joke of this form is one I made up myself: limit_(2->3) 2^2=37 I happen to like it a lot. === Subject: Re: Joke on limits I understand that larger value of 1 part, but I don't get the limits part. And therefore I don't understand your joke either, Qurqirish Dragon. Could somebody explain that part to me? type it before. But next time I'll make sure I'll use it. === Subject: Re: Joke on limits part. The first joke merely depends on treating 8 as a variable, and is essentially the same as the sufficiently large values of one joke. The second is a visual joke, which is why your presentation was commented on. A related limit would be Lim {w->3} x = x. > And therefore I don't understand your joke either, Qurqirish Dragon. > Could somebody explain that part to me? > type it before. But next time I'll make sure I'll use it. === Subject: Re: Extension of the Euclidean Metric to Subsets > <20128455.1126800027129.JavaMail.jakarta@nitrogen.mathforum.org>, Maury >>the problem which I'm thinking about is the following: >>let P(R^n) the power set of R^n (that is the set of all the subsets of R^n). >>Does there exist a metric m on P(R^n) such that for every x,y in R^n we have >>m({x},{y})=d(x,y), where d is the usual euclidean metric of R^n? >>For compact subsets we have the Hausdorff metric, which has this property. >>For all subsets the answer is negative, I think. >>What is your opinion? >>Maury > Sure, there is such a metric. For example, map P(R^n) bijectively onto R^(n+1) mapping > singletons {x} to (x,0) adding a single zero, and all other > sets bijectively onto the rest of R^(n+1). > Then use the usual metric on R^(n+1) to define your > metric on P(R^n). Not a pretty metric, but it satisfies your conditions. Surely you're not saying that the power set P(R^n) has the > same cardinality as R^(n+1)? I mean, aren't R^n and R^(n+1) > of the same cardinality (that being c)? OK, you're right. Map it onto some giant Hilbert space, with n coordinates reserved for the singletons, the rest any way you like. Better? -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Extension of the Euclidean Metric to Subsets <150920051518427635%edgar@math.ohio-state.edu.invalid> <160920050740205673%edgar@math.ohio-state.edu.invalid <20128455.1126800027129.JavaMail.jakarta@nitrogen.mathforum.org>, Maury >>the problem which I'm thinking about is the following: >>let P(R^n) the power set of R^n (that is the set of all the subsets of R^n). >>Does there exist a metric m on P(R^n) such that for every x,y in R^n we have >>m({x},{y})=d(x,y), where d is the usual euclidean metric of R^n? >>For compact subsets we have the Hausdorff metric, which has this property. >>For all subsets the answer is negative, I think. >>What is your opinion? >>Maury > Sure, there is such a metric. For example, map P(R^n) bijectively onto R^(n+1) mapping > singletons {x} to (x,0) adding a single zero, and all other > sets bijectively onto the rest of R^(n+1). > Then use the usual metric on R^(n+1) to define your > metric on P(R^n). Not a pretty metric, but it satisfies your conditions. Surely you're not saying that the power set P(R^n) has the > same cardinality as R^(n+1)? I mean, aren't R^n and R^(n+1) > of the same cardinality (that being c)? > OK, you're right. Map it onto some giant Hilbert space, with > n coordinates reserved for the singletons, the rest any way you like. > Better? > -- > G. A. Edgar http://www.math.ohio-state.edu/~edgar/ Your idea works for any infinite metric space. If (A,d) is a metric space let d^* be the discrete metric on P(A) where distinct elements all stand at distance 1 from each other. Then A x P(A) can be given the sup metric, d_s. There is clearly a bijection, f, from P(A) to A x P(A) which sends a singleton {a} to . Then d^**(B,C) = d_s(f(B),f(C)) works. A more interesting question is whether you can extend the euclidean metric to the space of all subsets in an interesting way. For example, union is continuous on the space of compact subsets with the Hausdorff metric. Can the continuity of union be preserved by an extension to all subsets? -John Coleman === Subject: Re: Extension of the Euclidean Metric to Subsets Sorry, Edgar, but this is a howler!!! A famous Theorem proved by Cantor states that, for any set S, S and P(S) are not equipotent. If you add that R^n and R^m have the same cardinality ... Maury === Subject: Re: Does this integral havea closed-form solution? > As for whether the erf is elementary, that's a matter of definition. > It's not usually taken to be so. And elementary as defined in the technical sense excludes erf. http://en.wikipedia.org/wiki/Elementary_function_(differential_algebra) -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Does this integral havea closed-form solution? > And elementary as defined in the technical sense excludes erf. > http://en.wikipedia.org/wiki/Elementary_function_(differential_algebra) Is there an algorithm for telling whether the indefinite integral of an elementary function is elementary? === Subject: Re: Does this integral have a closed-form solution? > And elementary as defined in the technical sense excludes erf. > http://en.wikipedia.org/wiki/Elementary_function_(differential_algebra) > Is there an algorithm for telling whether the indefinite > integral of an elementary function is elementary? This is known as Risch's algorithm. Part of it can be found in the book M. Bronstein, _Symbolic Integration I: Transcendental Functions_ (Springer-Verlag 1997) A lot more can be found in the archives of this newsgroup in threads on integration in finite terms and such. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Does this integral have a closed-form solution? >>Is there an algorithm for telling whether the indefinite >>integral of an elementary function is elementary? > This is known as Risch's algorithm. Part of it can be found in the book > M. Bronstein, _Symbolic Integration I: Transcendental Functions_ > (Springer-Verlag 1997) > A lot more can be found in the archives of this newsgroup in > threads on integration in finite terms and such. === Subject: Testable Predictions by HdB > I don't need any credits from you, Jesse. I will choose my own example, > preferrably one that you don't know anything about. So I'm not going to > fight in _your_ arena, with _your_ public applauding all over the place. > if you want to change the subject, by all means, give it your best > shot. > But explicitness surely helps. As I said before, one should expect > that when you give an example of a testable prediction, you include a > discussion of how to test these predictions and under what condition > the prediction is verified and/or falsified. Didn't take that conventional example sqrt(2) is irrational, because I wanted to do something interesting. And enjoyable for myself as well. Here is some Theory. Do not skip through. _Read_ it. It is only 4 pages long and it's not quite difficult: http://hdebruijn.soo.dto.tudelft.nl/jaar2005/italiaan.pdf Though it's not exactly my style, I included the common Theorem + Proof sequences and even some rudimentary set theory as well. :-) I have developed theory and practice from scratch within a week or so and would be glad to know if the material in this writeup contains any new insights. If not so: where to find references to the same stuff? Links to Fibonacci material are abundant even on the web. So it's very much probable, indeed, that my discoveries are not new altogether. Maybe some of you are wondering _why_ I find this stuff interesting. Well, it's a rather far-fetched purification of how to solve large sets of linear equations resulting from a finite element method for regular meshes if such meshes become infinitely large ... :-) A thought: the hand of God is in tiny details, not in generalities. Here comes a computer program, which is the implementation of several Numerical Experiments, associated with the above Theory, of course: http://hdebruijn.soo.dto.tudelft.nl/jaar2005/italiaan.zip Directions for use (Windows console application): - Make a subdirectory & go in there - Unzip (i.e. with pkunzip -d italiaan) - Run iterate(.exe = executable) - Follow instructions: 'iterate number' Now, what are the Testable Predictions of my Theory for the numerical experiments to be carried out with the program? 1. For values of 'number' equal to the Fibonacci Fractions F(k+1)/F(k) the iterands will become ill-conditioned. This is seen as 'oo' in the output (by catching the exception of a zero denominator). Suggested experiments: iterate 1 ; iterate 2 ; iterate 1.5 ; iterate 1.66666666666 ; iterate 1.6 . 2. The 'number' values (1+sqrt(5))/2 and (1-sqrt(5))/2 are invariants. This is seen as lines of output which should be all the same. Hint: you can copy and paste these values from 'iterate' without a numeric parameter. There is a difference, though. See below at (6). 3. Only numbers in the interval 1 < number < 2 which are not Fibonacci Fractions can result in iterands which remain positive for a while. Preferrably we should take an irrational number such as sqrt(2,3), indeed resulting, on output, in iterands which behave as described. Hint: you can copy and paste these values from 'iterate' without a numeric parameter. 4. Any number that is negative results in iterands which are negative as well. Any number < 1 results in negative iterands. A number > 2 results in one positive iterand < 1 and negative iterands for all the rest. 5. As iterands become negative, they will oscillate around the value (1-sqrt(5))/2 . You can check this with all numbers, except those at (1), as well as the value (1+sqrt(5))/2 . 6. More subtle. There is a difference if it comes to the invariants. The value (1-sqrt(5))/2 is _stable_ while the value (1+sqrt(5))/2 is _unstable_. The latter is worked out in a separate (and tricky) experiment called 'paradox'. Output shows that numbers are slowly shifting apart from (1+sqrt(5))/2 and converge to (1-sqrt(5))/2 in the end, nevertheless. Just run 'paradox'. But maybe I didn't tell you everything. Read the paper at 'Paradox?'. Well, Jesse, does the above fit the bill? Or does it not? Han de Bruijn === Subject: Re: Functions Distribution through any means other than regular usenet channels is forbidden. It is forbidden to publish this Content-Language: en >>Note: The author of this message requested that it not be archived. >>This message will be removed from Groups in 6 days (Sep 22, 3:57 am). Actually, I did not write this. Still you precede this >Since I'm responding to the post, his attempts have failed. === Subject: Re: Poll: Math and Calculators (was: How to Fix Texas Instruments) Just because some one did the orginal math to program the computer does not mean that the user knows the math. Also you have to know what the answer really means, is 8.4E-14 a true value or just the computers approximation for 0. ( had that happen once) hjs before you people finish teaching, you will be teaching physics to 3rd graders, who will be punching in differential equations in their hand held computers Dr. Willard J. Poppy during a lecture in Acoustics and Optics fall semester 1965. === Subject: Re: Why sci.math? (math or maths) <87slwe62w8.fsf@phiwumbda.org> <87fyse3xhl.fsf@phiwumbda.org> <87ll252zlv.fsf@phiwumbda.org> <87slwd11a7.fsf@phiwumbda.org> I have never stated what is, or ought to be, in Britain. You are revealing that to be your frame of reference. And when you put MY point of view, that math is correct and NOT a misspelling, into YOUR frame of reference you attributed to me a point of view that is not mine. Similarly, your frame of reference does not accommodate my proofs. Your subject bores me and I do not wish to be troubled. that I was contending with a Brit's goddamn nonsensical notion that math is a misspelling. I don't understand why you felt a need to contend with me. Kindly excuse my tardiness in correcting you. In the past six days I have given up 10 hours of leisure to working for comp time to be taken later, plus 5 hours of leisure to attending a wedding when I might otherwise have been working for comp time. David Ames > [Re: David's claim that maths is plural in British English] === Subject: Re: Why sci.math? (math or maths) <87slwe62w8.fsf@phiwumbda.org> <87fyse3xhl.fsf@phiwumbda.org> <87ll252zlv.fsf@phiwumbda.org> <87slwd11a7.fsf@phiwumbda.org I have never stated what is, or ought to be, in Britain. You are > revealing that to be your frame of reference. And when you put MY > point of view, that math is correct and NOT a misspelling, into YOUR > frame of reference you attributed to me a point of view that is not > mine. Similarly, your frame of reference does not accommodate my > proofs. Your subject bores me and I do not wish to be troubled. Your claims are just more ludicrous each day. You evidently know that the standard UK shortening of mathematics is maths and you claimed the s is evidence that it is plural. This is simply unsupported. So now, evidently, you have decided that you never meant to comment on UK usage. > that I was contending with a Brit's goddamn nonsensical notion that > math is a misspelling. I don't understand why you felt a need to > contend with me. > Kindly excuse my tardiness in correcting you. You're utterly adorable. I'm humbly chastised, I tell you what. -- But remember, as long as one human being follows the rules of mathematics, then mathematics as a human discipline survives. Right now I'm that one human being, so mathematics survives. -- James S. Harris === Subject: Re: Why sci.math? (math or maths) !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > Kindly excuse my tardiness in correcting you. In the past six days > I have given up 10 hours of leisure to working for comp time to be > taken later, plus 5 hours of leisure to attending a wedding when I > might otherwise have been working for comp time. Oh come on. Those 5 hours for getting married are bound to be a good investment. You'll save more than that alone from not having to match socks in future. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: open map On 15 Sep 2005 16:00:19 -0700, singau I meant the function >f: C^2 ---> C > (u,v) |--> u^2 Took me a second to see your point. Oh. Your point is that Jannick assumed that the function had a non-vabishing derivative in some direction, and when you said what about a local maximum? you really meant what about a point where the derivative in every direction vanishes? Think about it. We need to show that if O is open then f(O) is open. If there is _some_ point of O where the derivative in some direction is non-zero we're set. On the other hand if the derivative in every direction vanishes at every point of O (and O is connected) then f is constant. >But I didn't want to use z1 and z2 (we'd have z2^2, which is >cumbersome). Now the function is holomorphic, as it is analytic. >I'll see my prof next monday, so I'll ask him for the right answers, >but I was hoping there was a simple argument. >cheers, >singau ************************ === Subject: Re: open map On Fri, 16 Sep 2005 08:29:47 -0500, >On 15 Sep 2005 16:00:19 -0700, singau Hi Dale >>I meant the function >>f: C^2 ---> C >> (u,v) |--> u^2 >Took me a second to see your point. >Oh. Your point is that Jannick assumed that the function >had a non-vabishing derivative in some direction, and >when you said what about a local maximum? you really >meant what about a point where the derivative in >every direction vanishes? >Think about it. We need to show that if O is open then >f(O) is open. If there is _some_ point of O where the >derivative in some direction is non-zero we're set. >On the other hand if the derivative in every direction >vanishes at every point of O (and O is connected) then >f is constant. Oops. That only shows that f(O) has nonempty interior, sorry. Ok. It really is immediate from the case n = 1. Forget about those derivatives: Say O is a ball centered at c. Say L is the intersection of O with a complex line passing through c. If the restriction of f to L is non-constant then the case n = 1 shows that f(L) is open, and hence f(O) contains a neighborhood of f(c), which is what we need to prove. On the other hand if the restriction of f to L is constant for _every_ L then f is constant in O. >>But I didn't want to use z1 and z2 (we'd have z2^2, which is >>cumbersome). Now the function is holomorphic, as it is analytic. >>I'll see my prof next monday, so I'll ask him for the right answers, >>but I was hoping there was a simple argument. >>cheers, >>singau >************************ > ************************ === Subject: Re: open map <3oqmptF77j2pU2@individual.net> <4329661E.6040404@web.de> <61ili11l7ka77hbd9lpi204tubemd9v4rj@4ax.com> === Subject: Re: injective polynomial f: k x k -> k >On Thu, 15 Sep 2005 09:13:59 -0500, >>On Thu, 15 Sep 2005 15:03:56 +0200, Elias Vicari >Is it true? >For every field k (maybe only those of characteristic 0), it does not >exist a *injective* polynomial >f: k x k -> k >It seems true for k=R and k=Q, but what about in general? >>Maybe you don't really mean injective here? >He really does mean injective. >Elias asks: Is it true that there does _not_ exist an injective >polynomial f: k x k -> k? Indeed - I missed the not. >>If you >>mean what the word usually means then this is obviously >>false if k is finite, >Taking into account the above correction, we can say instead: > It's obviously true when k is finite. >>and it's also obviously false for >>k = R >which should be corrected to: > it's obviously true for k = R. >If f is in R[x,y] and f is 1-1, then f must be non-constant, so the >range of f is an interval (in R), possibly unbounded. If we consider >the graph of z=f(x,y) as a surface, then it seems clear that there are >at most finitely many values of z such that the f^(-1)(z) is a unique >point (x,y). In other words, the level curves of f cannot all >degenerate to single point sets. But this requires proof. I would >expect that there should be a simple, elegant proof using at most >calculus. >>(and it seems probably also obviously false for >>k = Q, I think.) >Again, the corrected version of your claim is: > (and it seems probably also obviously true for >k = Q, I think.) >But this is far from obvious. The very fact that you had to use the >qualifications it seems and I think makes it clear that it wasn't >so obvious to you. Perhaps you felt it was intuitively true, but I >know (and you know) that you couldn't immediately see why. >So now you have to prove it -- good luck. >Another case where the truth really is obvious is when k is >algebraically closed, since if f is identically 0, then f is clearly >not 1-1, and if f is not zero, the polynomial f(x,y)=0 has infinitely >many solutions, so again, f is not 1-1. >The fact that R is almost algebraically closed may offer a way to >prove the truth for k=R without resorting to calculus. >But Q is far from algebraically closed, and the pigeonhole principle >that works so easily for k finite doesn't directly apply for k=Q >(although indirectly, maybe it still does). >If we interpret the claim in terms of a diophantine equation in 4 >variables x1,y1,x2,y2, what it says is > The equation f(x1,y1)=f(x2,y2) always has solutions in Q other than > the trivial class of solutions x2=x1, y2=y1. >This makes the truth even less obvious (to me), so I would forget the >interpretation as a diphantine equation. >But to try to get back to a use of the pigeonhole principle, here's a >the possible line of attack for the proof that seems like it may work. >The idea is that polynomials in Z[x,y] can't increase fast enough in >all directions to avoid duplicating output values. >So let f be in Q[x,y] and suppose f is 1-1. Then d*f is also 1-1 for >any nonzero constant d in Q, hence, without loss of generality, we can >assume f is in Z[x,y]. The goal then is to try to find a region S in >Z^2 such that a < f(x,y) points in S is at least b-a. Then by the pigeonhole principle, f >cannot be 1-1 on S. >I have no time right now to try the above idea, and I'm not really >sure it can be made to work, but if it works for Q, then I sense that >the same idea can be adapted to apply to an arbitrary field k. >To explain the above intuition, suppose that for some field k, there >is a 1-1 polynomial f in k[x,y]. Then we can view f as a polynomial in >k'[x,y] where k' is the smallest field containing the prime field of k >together with all the coefficients of f. But now the field k' is >countable and f (viewed now as an element of k'[x,y]) is still 1-1. >So if we can prove the claim for all countable fields, then it's true >for all fields. But for countable fields, there may be a way to invoke >the pigeonhole principle as suggested above for k=Q. >quasi ************************ === Subject: Re: injective polynomial f: k x k -> k On Thu, 15 Sep 2005 15:26:57 +0100, Robert Low On Thu, 15 Sep 2005 15:03:56 +0200, Elias Vicari >Is it true? >For every field k (maybe only those of characteristic 0), it does not >exist a *injective* polynomial >f: k x k -> k >> Maybe you don't really mean injective here? If you >> mean what the word usually means then this is obviously >> false if k is finite, >It is obviously true when k is finite, because you >can't have any injective function at all from k x k to k. >At least, I'm reading his claim as 'for all k, there >is no injective polynomial k x k -> k' i.e 'for no k >is there an injective polynomial k x k -> k'. I missed the word not in the OP. ************************ === Subject: Re: injective polynomial f: k x k -> k > Is it true? > For every field k (maybe only those of characteristic 0), it does not > exist a *injective* polynomial > f: k x k -> k > It seems true for k=R and k=Q, but what about in general? A couple of remarks. First of all thank you for your participation. Quasi: > Another case where the truth really is obvious is when k is > algebraically closed, since if f is constant, then f is clearly not > 1-1, and if f is not constant, the polynomial f(x,y)=0 has infinitely > many solutions, so again, f is not 1-1. 1. I am not sure that you are right. f=0 it is known to have infinitely many solutions (on k closed) only when f is homogeneous (isn't it? maybe I am wrong). The case k closed is however solved in this way: on a paper by Bailynicki-Birula and Rosenlicht there is a 5-line proof of: if f: k^n -> k^n is an injective polynomial map, where k is algebraically closed field, then f is also surjective. (nice result itself) Using this theorem, one gets a contradiction assuming f: k^2 -> k injective and composing it with a non surjective inclusion i: k->k^2. 2. For the cases k=R,Q I also had a proof in the same spirit as a couple of yours, namely consider what happens with the image of a closed curve in R^2 or Q^2 (injectivity forces a contradiction to continuity). This is true for all continuous functions and also in particular for polynomials. Unfortunately such a proof needs the concept of continuity (also either you need a metric or a topology), which you cannot use for a general field. 3. Which structural property of a field rules out the existence of such a function? For example on N (set of natural numbers), there is such a function: f(a,b) = (a+b)^2 +3a + b (if you multiply by 1/2, you also get a bijection, but 1/2 is not a natural number...). What can be said about Z? === Subject: Re: injective polynomial f: k x k -> k >> Is it true? >> For every field k (maybe only those of characteristic 0), it does not >> exist a *injective* polynomial >> f: k x k -> k >> It seems true for k=R and k=Q, but what about in general? >A couple of remarks. First of all thank you for your participation. >Quasi: > Another case where the truth really is obvious is when k is > algebraically closed, since if f is constant, then f is clearly not > 1-1, and if f is not constant, the polynomial f(x,y)=0 has infinitely > many solutions, so again, f is not 1-1. >1. I am not sure that you are right. f=0 it is known to have infinitely >many solutions (on k closed) only when f is homogeneous (isn't it? maybe >I am wrong). Suppose f(X,Y) = sum_{i,j} c_{i,j} X^i Y^j is not constant. Let c_{m,n} be a nonzero coefficient. For any y in k, f(X,y) is a polynomial where the coefficient of X^m is sum_j c_{m,j} y^j. That's a non-constant polynomial in y, so it's nonzero for all but finitely many y. And when it's nonzero, f(X,y) is a non-constant polynomial so it has at least one root if k is closed. Since a closed field is infinite, we get infinitely many solutions of f=0 whenever k is closed. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: injective polynomial f: k x k -> k On Fri, 16 Sep 2005 09:24:51 +0200, Elias Vicari >> Is it true? >> For every field k (maybe only those of characteristic 0), it does not >> exist a *injective* polynomial >> f: k x k -> k >> It seems true for k=R and k=Q, but what about in general? >A couple of remarks. First of all thank you for your participation. >Quasi: > Another case where the truth really is obvious is when k is > algebraically closed, since if f is constant, then f is clearly not > 1-1, and if f is not constant, the polynomial f(x,y)=0 has infinitely > many solutions, so again, f is not 1-1. >1. I am not sure that you are right. f=0 it is known to have infinitely >many solutions (on k closed) only when f is homogeneous (isn't it? maybe >I am wrong). Hmm. I have not time now -- I have to rush out, but I'm fairly certain that for nonconstant f over an algebraically closed field k, f=0 will have infinitely many solutions. Here is a quick sketch of the proof (but maybe it's wrong): Choose any (homogeneous) form of f, call it g. There are only finitely many ratios y/x which make g identically 0, so by substituting y=tx there are only finitely many values of t which make g identically 0. For any other value of t, the form g survives, hence f survives, so there is at least one solution to f(x,tx)=0. But since there are infinitely many possible t's to choose from, we get infinitely many ratios y/x among the solutions, hence infinitely many solutions. >The case k closed is however solved in this way: on a paper >by Bailynicki-Birula and Rosenlicht there is a 5-line proof of: if f: >k^n -> k^n is an injective polynomial map, where k is algebraically >closed field, then f is also surjective. (nice result itself) >Using this theorem, one gets a contradiction assuming f: k^2 -> k >injective and composing it with a non surjective inclusion i: k->k^2. >2. For the cases k=R,Q I also had a proof in the same spirit as a couple >of yours, namely consider what happens with the image of a closed curve >in R^2 or Q^2 (injectivity forces a contradiction to continuity). Can you show the proof for k=Q? quasi === Subject: Re: injective polynomial f: k x k -> k > Can you show the proof for k=Q? > quasi Hum, I am sketching the proof for Q, without any epsilon-delta details ;) Choose a closed curve c in Q^2. If you have problem to figure out a closed curve in Q^2 (as I do), consider the zero set of x^2 + y^2 = 1 (since Q is dense in R you are fine). Choose a point P on c and consider its image on Q. If you start moving along c, your images keeps moving in one direction on Q (monotonicity). Of course, once you approach P again, you are far from the original image of P, contradicting the epsilon-delta formulation of continuity. As I said, this proof has nothing to do with algebra and therefore is by no way applicable to an abstract field k. === Subject: Re: injective polynomial f: k x k -> k <9tnli1lmfkjcgmm3t1mpv4outsitu56i0n@4ax.com> <432ad6da$1@news1.ethz.ch Can you show the proof for k=Q? > quasi > Hum, I am sketching the proof for Q, without any epsilon-delta details ;) > Choose a closed curve c in Q^2. If you have problem to figure out a > closed curve in Q^2 (as I do), consider the zero set of x^2 + y^2 = 1 > (since Q is dense in R you are fine). Choose a point P on c and consider > its image on Q. If you start moving along c, your images keeps moving in > one direction on Q (monotonicity). Of course, once you approach P again, > you are far from the original image of P, contradicting the > epsilon-delta formulation of continuity. Sorry, I don't see the monotonicity part. A continuous injective function from Q to Q doesn't have to be monotonic. In fact, there are entire functions that map Q injectively into Q, and are not monotonic. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: (2^(1/2))^(1/2)......=1 > By using calculator of 9 decimal digits I get the repeated root of 2 equal > 1 after 32 step > Now if the calculator has n decimal digits, after how many steps shall I > get 1 x_n = a^(1/b^n) we are trying to find x_n - x_(n-1) < e or a^(1/b^n) - a^(1/b^(n-1)) < e a^(1/b^n) - a^(b/b^n) < e let y = a^(1/b^n) then we have y - y^b < e let b=2 in our case for simplicity and lets solve y^2 - y + e = 0 which has solution y = (1 +- sqrt(1 - 4e))/2 or a^(1/2^n) = (1 +- sqrt(1 - 4e))/2 or 1/2^n = log[a]((1 +- sqrt(1 - 4e))/2) n = -log[2](log[a]((1 +- sqrt(1 - 4e))/2)) n = ceil(-log[2](log[a]((1 +- sqrt(1 - 4/10^m))/2))); where m is the mth decimal place you want to find n for. Jon === Subject: Re: (2^(1/2))^(1/2)......=1 > athforum.org... > By using calculator of 9 decimal digits I get the > repeated root of 2 equal > 1 after 32 step > Now if the calculator has n decimal digits, after > how many steps shall I > get 1 > x_n = a^(1/b^n) > we are trying to find x_n - x_(n-1) < e It should x_(n-1) - x_n < e > or > a^(1/b^n) - a^(1/b^(n-1)) < e > a^(1/b^n) - a^(b/b^n) < e > let y = a^(1/b^n) > then > we have y - y^b < e > let b=2 in our case for simplicity and lets solve If b=11 !!!! > y^2 - y + e = 0 > which has solution > y = (1 +- sqrt(1 - 4e))/2 > or > a^(1/2^n) = (1 +- sqrt(1 - 4e))/2 > or > 1/2^n = log[a]((1 +- sqrt(1 - 4e))/2) It should 1/2^n = log((1 +- sqrt(1 - 4e))/2)/log(a) === Subject: Re: (2^(1/2))^(1/2)......=1 >> athforum.org... >> By using calculator of 9 decimal digits I get the >> repeated root of 2 equal >> 1 after 32 step >> Now if the calculator has n decimal digits, after >> how many steps shall I >> get 1 >> x_n = a^(1/b^n) >> we are trying to find x_n - x_(n-1) < e > It should x_(n-1) - x_n < e yes... >> or >> a^(1/b^n) - a^(1/b^(n-1)) < e >> a^(1/b^n) - a^(b/b^n) < e >> let y = a^(1/b^n) >> then >> we have y - y^b < e >> let b=2 in our case for simplicity and lets solve > If b=11 !!!! then you solve the corresponding 11th degree polynomial. Those polynomials of that form I think, but I'm not sure, are solvable. The point is you asked how to do it and I provided a general method... I'm not going to work out the details of an 11th degree polynomial. >> y^2 - y + e = 0 >> which has solution >> y = (1 +- sqrt(1 - 4e))/2 >> or >> a^(1/2^n) = (1 +- sqrt(1 - 4e))/2 >> or >> 1/2^n = log[a]((1 +- sqrt(1 - 4e))/2) > It should 1/2^n = log((1 +- sqrt(1 - 4e))/2)/log(a) doesn't matter, it is the same... log[a](x) = log(x)/log(a). Its called the change of base formula the main thing to fix is the sign... but as you see it really didn't effect anything. (i.e. just use -e instead of e because I sovled the equation and not the inequality). === Subject: Re: (2^(1/2))^(1/2)......=1 An upper limit for the number of iterations required can be trivially found by observing that the square-root of 1+x is less than 1+x/2. Thus, log_2(10^n)+1 will work for n digits (the +1 is so that the first non-displayed digit will be under 5, and therefore rounding will set the display to 1.) === Subject: Re: (2^(1/2))^(1/2)......=1 It 's just an iteration problem. We can count the iteration number of a function f(x) = x^p , p any real positive with a phi(x) Abel function : phi(x) = ln(ln(x))/ln(p) + c , c a constant. Here p = 1/2 ; Remains to adjust c to your given steps: for step n , n = (ln(ln(2)) - ln(ln(x_n) )/ln(2) + 1 , x_n the corresponding value of n th step ; come on , Alain. === Subject: Re: Kinghts & Knaves <4324ECFB.5000301@it.com> <43278B04.C47E4F5E@worldnet.att.net> [... -- Knights always tell the truth, Knaves always lie] >> In the version I've read, that was taken care of in the following way: the >> traveller asks two people, but can't hear or understand the first one's >> answer; so the second explains: He said he's a knave. I'm a knight. >> Since, for the same reasons as you give, no one in that society would ever >> say that he's a knave, the second person must be lying and must therefore >> himself be a knave. > But someone could say: He said he's a knave, AND I'm a knave. That means > the person is a knave (he/she couldn't be a knight), so the compound > statement is false. The second part (I'm a knave) is true, so the first > part (He said he's a knave) is false, so the other person did not say he > was a knave. You're trying to be a little too smart by half. Since this is just a logic problem and not a real world example, the problem would have to be properly framed. We are told NOTHING about compound questions and rules regarding them. To complicate it like this is essentially meaningless. > But not saying you're a knave is not the same as saying you're not a knave, > so that's as far as you can go. (The mumbling could have been unrelated to > the question.) I could have objected to his example by saying, what happens if the second person is a Knight? The problem wouldn't work and thus no one would ever frame it that way, so it's pointless to even ask such a question. -- Theodore A. Kaldis kaldis@worldnet.att.net === Subject: Re: Probability Theory Is Inconsistent > Of course, ALL applications of deductive mathematics are subject to the > little proviso that certain assumptions are made and that physical > events are subject to the laws of ... Certainly people didn't used to > think like this; magic has been prevalent as explanation for far more > thousands of years than logic. But physical events are subject to the laws of probability is not logic. It's only an empirical law which is subject to modification in the face of new evidence. An apparently purposeful world is just as reasonable a priori as the apparently purposeless (random) one we find empirically. This needn't even involve anything resembling human intelligence: for example, there are models with closed causal loops in which events conspire in a seemingly purposeful way to prevent inconsistency. This happens for much the same reason that Deep Blue seems to understand chess. A magical world might still be amenable to logical analysis. > I suspect the belief in such magic is still widespread (else WHY > would ANYONE go to Las Vegas?). Because I have relatives there. But seriously, why does anyone go to movies? You pay $10 to get in the building, and a few hours later you walk out with nothing but memories. People who gamble pay money in exchange for a fantasy of striking it rich. It's not fundamentally different from any other form of entertainment. If you think that gambling is irrational, you're using the wrong utility function. (Of course, there are many gamblers who exhibit addictive behavior, but that's a separate issue.) -- Ben === Subject: Re: Probability Theory Is Inconsistent <120920050733577733%edgar@math.ohio-state.edu.invalid> [Find it in his Miscellany. It is worth reading.] > http://www.edge.org/q2005/q05_8.html#susskind > -- Ben This thread proved only ONE thing -- that a bunch of supposedly education grown-ups in mathematics, statistics, and probability gets all tangled on two ENTIRELY DIFFERENT questions: 1. The ASSESSMENT of a uncertain number p, called probability. 2. The consistency of probability LAWS governing assessed p's. Take a timely question in which it is meaningful to ask a probability question: Q. What's the probability that Delta airlines will be able to avoid liquidation under the Chapter 11 filing, to emerge as a non-bankrupt operating airline out of Chapter 11, say, within two years? Obviously there is no FREQUENTISTIC interpretation of this p, because nothing is repeatable, for one thing. It is a one- time affair that has a measure of UNCERTAINTY, and everyone has his/her OPINION (personal assessment) of that probability, and there are no right or wrong answer until after the event is finished, checking only hindsigh after the 1-time trisl. What about probability THEORY and probability LAWS? That's an entirely separate and different question altogether. For example, if the probability of success is assessed to be p, then the probability of failure or no success is (1 - p), no matter WHAT the assessed value of p is. It's inconsistent only if Joe BLow says his p is 0.25 and his assessment of the complement of that event is 0.65, then Joe Blow has a defective probability space, which is not consistent in the sense of the word being tossed around in the subject and the discussion which is off the mark. There are numerous other ways for the consistent set of probability laws to be violated, but NONE of it is because someone gave an incorrect assessment of a single probability, in and by itself, such as the probability of flipping coin and getting a HEAD. You know the other probability LAWS. They are consistent, coherent, and have perfectly good OPERATIONAL meanings even though some events have very different assessments of p by various experts. That's what makes one investor a better one another, in all walks of BUSINESS. So, forget about the meaning of probability from a frequentist (and repeatable event) point of view and forget about all that convergence and other technical irrelevance on probability theory itself, and concentrate on the subjective meaning of probability, such as What's the PROBABILITY that a share of DAL (Delta stock) will rise above $1 by the end of this month)? To understand the process of probability elicitation and assessment, let's consider the following game (which is an obvious adaptation of serious probability assessment ideas), such as those found in many references contained in my post in this thread, on which no-one seemed to have paid due attention because of their own confusion about items (1) and (2) at the beginning of this post. Read the meaning of subjective probability and the hundreds of references embedded in the two references the post a few days ago: http://tinyurl.com/8mdsv Below is one of the many different ways of assessing one's PERSONAL probability about any event. For the sake of simplicity, I'll use that as an illustration of how anyone can assess his own probability of the probability of a HEAD on one toss of a fair (or unfair, for that matter) coin. You can assess YOUR assessed probability of ANY future event using the same (conceptual, if not game-show-like) method. Have you heard of the TV show Who Wants to be a Millionaire? Here's a version of the game that can be understood by anyone with some sense rather than the physicists and mathematicians who make imaginary booby traps to trip themselves all over the place. Here's how the game is played. Instead of a series of questions you have to give the correct answers to advance toward the $M, you have a series of CHOICES between two alternatives which you think has a better chance or probability of winning the $M, for YOU. Instead of the three life lines when the contestent is not sure of the answer to the questions, to get help from others, you have three opportunities to say I am not sure which I prefer; or I am indifferent between the two choices. The game ends when you have used up your three life lines. Here are the two choices in each question: A. You win $1,000,000 if you flip a coin and it comes up HEADS. B. You win $1,000,000 if you draw a bead from a bag of 1,000,000 beads of X red ones and (1,000,000 - X) black ones, thoroughly mixed at random, and draws a red one. The game show host, Monte Philbin, tells you what X is each time, and you decide whether you prefer A to B, or vice versa. 1st question. X = 1000. Contestant has no problem choosing A as the final answer. The audience cheers on his wise choice. The questions get progressively harder -- in this case, to make the choice between A and B. 17th question: X = 495,000. Contestent had to think harder ... looks up the ceiling for inspiration, scratches his head as if it would help, and then chose A. 18th, 19th, and 20th questions, the X's were 499,900, 199850, and 499,950, and the Contestent could not make a clear choice between A and B, and the game was OVER. That means the Contestant has assessed his SUBJECTIVE probability of the probability of Heads of the coin to be between .499850 and .499950. The statistically untrained audience groans! They though he had come so close to the right answer and lost. But THEY were wrong. The Contestent CANNOT be wrong about his PERSONAL probability. If he has to pick a single number as his probability of HEADS in the toss of a single coin, that probability in the chosen interval of indifference would be CORRECT. That's the ASSESSMENT of the probability of an event that has a frequentistic right answer, by reason of symmetry or other necessarian views. The Contestant WON a door price from Monte Philbin for playing the game and got the right answer. Every Contestant wins in this show. Just because the show is named Who Want To be a Millionaire doesn't mean any contestant ever wins more than $10,000. In short, you can assess YOUR probability on any event to as many place decimal accuracy you are able to distinguish betwen two close choices. THEN, let the probability laws take over the operation on those assessed probabilities. They will be consistent. -- Bob. === Subject: Re: Probability Theory Is Inconsistent >Littlewood has an essay something like this. >This thread proved only ONE thing -- that a bunch of supposedly >education grown-ups in mathematics, statistics, and probability >gets all tangled on two ENTIRELY DIFFERENT questions: >1. The ASSESSMENT of a uncertain number p, called probability. >2. The consistency of probability LAWS governing assessed p's. I don't think I'm confusing these two questions, and I don't think Susskind or Littlewood were either. I think you're making the same mistake as the OP, which is assuming that people are unaware of something just because they happen not to mention it. I'm also not convinced that you understand the particular metaphysical difficulty that Susskind is writing about (and also Littlewood I assume, though I haven't read his essay yet). > You know the other probability LAWS. They are consistent, > coherent, and have perfectly good OPERATIONAL meanings even though > some events have very different assessments of p by various experts. > That's what makes one investor a better one another, in all walks > of BUSINESS. Careful: it sounds like you're claiming here that investors who use a good probability model will outperform investors who make investment decisions by consulting an astrologer. If so, the question is, will they outperform them in every investment, or just most of the time? Hint: don't say most of the time. Let me try to explain the problem in detail. As a Bayesian, you follow a certain procedure and obtain a number representing the subjective probability you assign to some event. Now, what do you actually do with these numbers? Presumably you act on them in some way -- for example, you work out a subjective projected profit from investing in stock A, and a subjective projected profit from investing in stock B, and invest in A or B according to which number is higher. Thus far I (playing the skeptic) have no problem with this. You're free to live your life in any way you choose, as long as it doesn't impinge on my freedoms, yada yada yada. The problem shows up if you try to convince *me* to become a Bayesian. I don't mean convince me to use your subjective probabilities, just to believe in the whole notion of subjective probability and Bayesian inference as a good way to make decisions. There are basically two ways to do this: 1. Appeal to prejudice: It obviously makes sense. 2. Appeal to evidence: It works. The problem is that argument 2 is frequentist (or circular), and argument 1 is not scientific. Possibly you believe that the internal self-consistency of Bayesian reasoning is enough to justify its application in practice. That is easily disposed of. Since any outcome of a probabilistic event is possible (i.e. obtains in some possible world), the actual world could be one in which probabilistic methods work badly. It could even be one in which probabilistic methods work well for people named Bob, and badly for everyone else. But it isn't; why not? Not only is the answer to this question unclear, it isn't even clear how to formulate the question in a rigorous way. That's the difficulty that Susskind is talking about. -- Ben === Subject: Re: Probability Theory Is Inconsistent <120920050733577733%edgar@math.ohio-state.edu.invalid> This thread proved only ONE thing -- that a bunch of supposedly >education grown-ups in mathematics, statistics, and probability >gets all tangled on two ENTIRELY DIFFERENT questions: >1. The ASSESSMENT of a uncertain number p, called probability. >2. The consistency of probability LAWS governing assessed p's. > I don't think I'm confusing these two questions, and I don't think Susskind > or Littlewood were either. If not, then they wouldn't be arguing about the question of INCONSISTENCY in Probability Theory on only one of the many meanings of Probability. None of the convergence, correctness of p, etc. apply to (1). > I think you're making the same mistake as the OP, > which is assuming that people are unaware of something just because they > happen not to mention it. Read my statement again, more carefully this time. I didn't assume anything about whether people are aware or unaware of subjective probabilities; only they were CONFUSESD about (1) and (2), which are different issues no matter how (1) is defined or treated, classical, frequentist, Bayesian, neoBayesian. (1) and (2) are SEPARATE issues, period! > I'm also not convinced that you understand the > particular metaphysical difficulty that Susskind is writing about (and also > Littlewood I assume, though I haven't read his essay yet). I got the full subtlety of Susskind's humor, call it metaphysical if you will. That is a horse of a THIRD different color -- hunor, sarcasm, satire, parady, and other forms of prose about technical subjects. > You know the other probability LAWS. They are consistent, > coherent, and have perfectly good OPERATIONAL meanings even though > some events have very different assessments of p by various experts. > That's what makes one investor a better one another, in all walks > of BUSINESS. > Careful: it sounds like you're claiming here that investors who use a good > probability model will outperform investors who make investment decisions by > consulting an astrologer. If so, the question is, will they outperform them > in every investment, or just most of the time? Hint: don't say most of the > time. Your bad. :-) It has nothing to do with any probability MODEL. It has to do with how good the ASSESSMENT of p is, as in (1). If you think an investment decision based on the assessment of the probability of success is like consulting an astrologer, you have much more to learn than just the difference between (1) and (2). When Yahoo stocks was first issued, I know enough about it to think of the probability of its success (say doubling within a year) was as better than 0.5 or flipping a coin and getting HEADS. That turned out to be a good assessment of p and a good investment when I bought a bunch on the 2nd day of issue -- whose price doubled several times within the first year. :-) There was no MODEL of any kind. p and a good gamble that paid off. > Let me try to explain the problem in detail. As a Bayesian, you follow a > certain procedure and obtain a number representing the subjective > probability you assign to some event. Now, what do you actually do with > these numbers? Presumably you act on them in some way -- for example, you > work out a subjective projected profit from investing in stock A, and a > subjective projected profit from investing in stock B, and invest in A or B > according to which number is higher. That's acceptable. > Thus far I (playing the skeptic) have no problem with this. You're free to > live your life in any way you choose, as long as it doesn't impinge on my > freedoms, yada yada yada. The problem shows up if you try to convince *me* > to become a Bayesian. I did no such! You can bring a horse to water, but you can't make it do a backstroke ... or something like that. You are free do whatever you do, including consulting your astrologer. You can also choose to believe all the frequentist and convergence stuff as NECESSARY before you can think coherently about probability and its applications. That ITSELF is your SUBJECTIVE probability assigned to the correctness of one over the other. You are a Bayesian whetehr you like is or not! > I don't mean convince me to use your subjective > probabilities, just to believe in the whole notion of subjective probability > and Bayesian inference as a good way to make decisions. There are basically > two ways to do this: > 1. Appeal to prejudice: It obviously makes sense. > 2. Appeal to evidence: It works. > The problem is that argument 2 is frequentist (or circular), and argument 1 > is not scientific. That's your own misguided notion about what's scientific and what's not, and the rest of your own muddled thinking about the subject of UNCERTAINTY, how to assess it, and how to apply what you assessed, coherently and consistently. > Possibly you believe that the internal self-consistency of Bayesian > reasoning is enough to justify its application in practice. Not at all! Read some of those references I gave about Bayesian probability and statistics. Being able to elicit your OWN probability assessment of whether Tennessee will beat Florida in the game tomorrow has nothing to do with whether Bayssian statistics is internally consistent or not. > That is easily > disposed of. Since any outcome of a probabilistic event is possible (i.e. > obtains in some possible world), the actual world could be one in which > probabilistic methods work badly. It could even be one in which > probabilistic methods work well for people named Bob, and badly for everyone > else. But it isn't; why not? Not only is the answer to this question > unclear, it isn't even clear how to formulate the question in a rigorous > way. That's the difficulty that Susskind is talking about. > -- Ben After all that mouth-dancing, are you saying it is NOT meaningful for anyone to assess the probability p that Florida will beat Tenn in the football game tomorrow (if you're foreign excuse the particular example in the USA)? How does a frequentist (or any other brand of probabilist) assess that probability p? -- Bob. === Subject: Re: Probability Theory Is Inconsistent <120920050733577733%edgar@math.ohio-state.edu.invalid> Here are the two choices in each question: >> A. You win $1,000,000 if you flip a coin and it comes up >> HEADS. >> B. You win $1,000,000 if you draw a bead from a bag of >> 1,000,000 beads of X red ones and (1,000,000 - X) black >> ones, thoroughly mixed at random, and draws a red one. >> The game show host, Monte Philbin, tells you what X is >> each time, and you decide whether you prefer A to B, >> or vice versa. I guess I am having trouble following your point here. It seems to me that the best chance of success is choosing A when X < 500,000 and choosing B when X > 500,000 (assuming a fair coin and a bead chosen with replacement). 18th, 19th, and 20th questions, the X's were 499,900, >> 199850, and 499,950, and the Contestent could not >> make a clear choice between A and B, and the game >> was OVER. Why again does the contestant not have a clear choice here? >> That means the Contestant has assessed his >> SUBJECTIVE probability of the probability of Heads >> of the coin to be between .499850 and .499950. Huh? I am definitely not following this. The phrase the probability of the probability is confusing to me. In a fair coin (barring that it lands on its side), the probability of it landing heads is 0.5. And I don't know what you mean by the subjective probability. >> The statistically untrained audience groans! They >> though he had come so close to the right answer >> and lost. But THEY were wrong. Wrong about which? What are you assuming they are groaning about? >> The Contestent CANNOT be wrong about his PERSONAL >> probability. If he has to pick a single number as his >> probability of HEADS in the toss of a single coin, that >> probability in the chosen interval of indifference would be >> CORRECT. I am not following your concept of personal probability either. A fair coin is going to come up heads with probability 0.5, irrespective of what is personal thoughts are (again, assuming a fair coin that doesn't land on edge). Perhaps there are other rules to this game that I did not notice? Jonathan Hoyle === Subject: svd of block matrices can anyone provide me a good reference about svd's of block matrices? I mean, what's the relation between sdvs of A and B and svd of [A;B] or [A B]? (where A and B are complex matrices) I would like to update the computation of singular vectors when rows or columns are added to a matrix ... jonagold, Belgium === Subject: Not orthogonal spherical cap Hello! I really need your help: I have some images of sperical caps, taken at a given vertical visual angle. I'd like to calculate the radius and the height by analysing the edges, but I really dont know how to do that in a sufficiently precise way. Can someone suggest me a webpage, a book or a paper where I can find this procedure? Marco === Subject: Re: Not orthogonal spherical cap Supersedes: [corrected -- I said radius when I meant diameter] >I really need your help: I have some images of sperical caps, taken at >a given vertical visual angle. >I'd like to calculate the radius and the height by analysing the edges, >but I really dont know how to do that in a sufficiently precise way. >Can someone suggest me a webpage, a book or a paper where I can find >this procedure? If your camera is far enough away from the cap that perspective is not an issue, I think it breaks down into 2 cases: 1. The entire edge of the cap is visible in the image. In this case, the shape is an ellipse, the same ellipse you would see if spherical cap were a flat circle. The shading will be different, but the edge will be the same. In this case you can find the diameter of the circle (it's equal to the major axis of the ellipse) but not the sphere. Since you know the angle, you can calculate a minimum diameter for the sphere but not a maximum, which gives you a maximum cap height but not a minimum. 2. Half of the edge is visible on one side, and the other side of the shape is a horizon of the sphere. Again, the longest axis is the diameter of the circular cross-section. The near half of the shape will be a half-ellipse, and the far side will be a circular arc. The diameter of the circular arc is the same as that of the sphere. Given the diameter of the circular cross-section and the diameter of the sphere, the height should be easy to calculate. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: I need help. abc=1,a>0,b>0,c>0. 1/a+1/b+1/c+3/(1/a+1/b+1/c)>=4. How to do it? === Subject: Re: I need help. > abc=1,a>0,b>0,c>0. > 1/a+1/b+1/c+3/(1/a+1/b+1/c)>=4. > How to do it? that means: bc + ac + ab + 3/(bc + ac + ab) >= 4 being x=bc + ac + ab>0 you can write x^2 -4x + 3 >=0 so you obtain x<=1 and x>=3 rewrite it in a,b,c terms. the first hypotesis says bc + ac + ab < 1 if u moltiply by a (you can, a is positive), and considering abc=1, and then dividing for a^2 you arrive at the disequation: c+b<= (a-1)/a^2 so a needs to be greater than 1 but if u solve in the same way moltypling by b etc etc you obtain that either b has to be greater than one, and than is impossible. Instead the other hyp (x>=3) is easly satisfied. working in the same way you obtain the condition a>=1/3, b>=1/3, c>=1/3 but c=1/ab so it means that, ab<=3 and again in the same way, ac<=3 and bc<=3 for example you can choose a=1/3 b=1/3 c=9 and your disequation is satisfied. === Subject: Re: I need help. > abc=1,a>0,b>0,c>0. > 1/a+1/b+1/c+3/(1/a+1/b+1/c)>=4. > How to do it? Are you allowed to use calculus? Because of the apparent symmetry among the variables a,b,c, intuition begs us to check the case a=b=c. So, let's try to verify that with logic. Let f(a,b,c) = 1/a+1/b+1/c+3/(1/a+1/b+1/c), and g(a,b,c) = abc. We want to find the infimum of f on the surface of g = 1. Then denote the gradient of f as grad f = (f_a, f_b, f_c) and we can easily calculate the gradient of g as grad g = (bc, ac, ab). Now, if we're on the surface of abc=1 and find that the two gradients are not equal, then we can move along the projection of grad f onto grad g to increase f. So, equating the gradients yields f_a = bc, f_b = ac, f_c = ab or a f_a = b f_b = c f_c. Now, just just look at any pair of variables, say a and b. If a f_b if f_a>0 or f_a < f_b if f_a<0. In either case, the gradients are not equal. So this shows that we cannot achieve the infimum as long as a,b,c are not all equal to each other. So, we conclude we must have the condition a = b = c. The rest is easy now. Now we have at the infimum that a = b = c = t, 1/a+1/b+1/c+3/(1/a+1/b+1/c) = 3/t + t. By the abc=1 constraint, we have t=1. So, the infimum is 3/t + t = 4. So, we have the result 1/a+1/b+1/c+3/(1/a+1/b+1/c) <= 4. Probably not very rigorous, but that's just details. :) -kira === Subject: two questions I have two problems now: 1. A theorem states: A necessary and sufficient condition for a complex matrix A to be positive definite is that the Hermitian part A* = 1/2 (A+A') be positive definite. My question is if it is much easier to see the positive definitinity of A* than A itself? 2. Theorem states: If X are a vector of random variables such that no element of X is a linear combination of remaining elements (that is, there don't exist a( a not qual to zero) and b such that a'*x=b for all values of X=x), then covariance matrix D(X) is a positive definite matrix. Here, I am wondering, if the element of X is independ variable to others with dimension of N. Then is that maximum number of elememnts of X should be N? === Subject: Re: two questions > I have two problems now: > 1. A theorem states: > A necessary and sufficient condition for a complex matrix A to be positive definite is that the Hermitian part > A* = 1/2 (A+A') > be positive definite. This is easily proven, since for any vector x, x'A*x = Re[x'Ax] > My question is if it is much easier to see the positive definitinity of A* than A itself? Often, yes, I think so, due to the fact that there are theorems about Hermitian matrices that can be brought to bear. > 2. Theorem states: > If X are a vector of random variables such that no element of X is a linear combination of remaining elements (that is, there don't exist a( a not qual to zero) and b such that a'*x=b for all values of X=x), then covariance matrix D(X) is a positive definite matrix. > Here, I am wondering, if the element of X is independ variable to others with dimension of N. Then is that maximum number of elememnts of X should be N? I don't understand what you are asking. X is a vector of random variables, either independent or not. It has N elements. That's some fixed, predefined value. - Randy === Subject: Cute little math puzzle I just bought a big jar of Jelly Belly jelly beans at Costco. There are 49 flavors. Assuming that there are equal numbers of each flavor, how many beans do I have to eat before it becomes likely that I've eaten two beans of the same flavor? There's actually 1.814 Kg of these beans in the jar and they say that 35 of them are about 40 grams. But for the sake of this puzzle, let's just assume it's a real big (infinite) vat of jelly beans, and you aren't diminishing the chance of pulling a watermelon flavored bean just because you already ate one. Note: To avoid any possible misunderstanding I will restate this in more mathematical terms. I often notice things like whether or not there are duplicate integers in a string composed of single digit integers. For instance: 209384 No duplicate integers 397505 Duplicate integers 349657 No duplicate integers 345760 No duplicate integers 559673 Duplicate integers Rephrasing the original problem in more mathematical terms: How many single digit integers do I have to pick in base 49 until I have probably picked two of the same. === Subject: Re: Cute little math puzzle >I just bought a big jar of Jelly Belly jelly beans at Costco. There are >49 flavors. Assuming that there are equal numbers of each flavor, how >many beans do I have to eat before it becomes likely that I've eaten two >beans of the same flavor? The probability that you have *not* selected two of same flavor in the first n beans 49! / [(49-n)! 49^n] for n < 50. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: Cute little math puzzle !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > I just bought a big jar of Jelly Belly jelly beans at Costco. There > are 49 flavors. Assuming that there are equal numbers of each > flavor, how many beans do I have to eat before it becomes likely > that I've eaten two beans of the same flavor? Depending on your definition of likely, the answer can be anything from 2 to 50. > Note: To avoid any possible misunderstanding I will restate this in more > mathematical terms. I often notice things like whether or not there are > duplicate integers in a string composed of single digit integers. For > instance: > 209384 No duplicate integers > 397505 Duplicate integers > 349657 No duplicate integers > 345760 No duplicate integers > 559673 Duplicate integers > Rephrasing the original problem in more mathematical terms: How many > single digit integers do I have to pick in base 49 until I have probably > picked two of the same. Depending on your definition of probably, the answer can be anything between 2 and 50. The probability that picking n beans does not yield a duplicate is n!*(49 choose n)/49^n So the probability for 0 beans not yielding a duplicate is 1, for 1 bean it is 1, for two beans it is 48/49, for three it is 47*48/49^2 and so on. The 50% point is reached at 9 beans, where the probability of at least one double becomes 0.54%. With 49 beans, you still have a probability of about 9*10^{-21} that all beans are unique. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cute little math puzzle : :> I just bought a big jar of Jelly Belly jelly beans at Costco. There :> are 49 flavors. Assuming that there are equal numbers of each :> flavor, how many beans do I have to eat before it becomes likely :> that I've eaten two beans of the same flavor? : :Depending on your definition of likely, the answer can be anything :from 2 to 50. : :> Note: To avoid any possible misunderstanding I will restate this in more :> mathematical terms. I often notice things like whether or not there are :> duplicate integers in a string composed of single digit integers. For :> instance: :> 209384 No duplicate integers :> 397505 Duplicate integers :> 349657 No duplicate integers :> 345760 No duplicate integers :> 559673 Duplicate integers :> Rephrasing the original problem in more mathematical terms: How many :> single digit integers do I have to pick in base 49 until I have probably :> picked two of the same. : :Depending on your definition of probably, the answer can be anything :between 2 and 50. : :The probability that picking n beans does not yield a duplicate is :n!*(49 choose n)/49^n : :So the probability for 0 beans not yielding a duplicate is 1, for 1 :bean it is 1, for two beans it is 48/49, for three it is 47*48/49^2 :and so on. : :The 50% point is reached at 9 beans, where the probability of at least :one double becomes 0.54%. : :With 49 beans, you still have a probability of about 9*10^{-21} that :all beans are unique. thread: http://forums.anandtech.com/messageview.aspx?catid=50&threadid=1690036&enter thread=y === Subject: Re: Infinity =/= Infinity > Let's have an infinite set Y consisting of elements. Cause ist's a set, > the elements are pairwise different. > Now take one element out of Y and establish a set A including just this > one element. Take two elements out of Y and establish set B with this > two elements, than take three elements to establish C, and so on. > Since we consider to do this infinitely many times in one step we can > have instantely a new set Z which elements are the sets A, B, C, ... . > So, every element of Z is a set and consists of elements. You're trying to describe an algorithm that copies all of the elements of Y, an infinite set, into set Z. Assuming this works, the resulting set Z is exactly the same set that Y was. > If there is such an infinite element in Z, there must be an infinite > number in N - which is impossible. So there can't be an infinite-valued element in Z. Was there such an element in Y at the start that got moved from Y into Z? The Peano axioms specifically allow for the existence of infinite sets. They do not allow for the existence of infinite naturals (or, more precisely, the Peano successor operation only produces finite naturals). What works for sets is entirely independent for what works for naturals. The fact that a set has the property of being infinite (i.e., it contains an infinite number of elements), does not say anything about the properties of the elements within the set themselves. === Subject: Re: Infinity =/= Infinity > Let's have an infinite set Y consisting of elements. Cause ist's a set, > the elements are pairwise different. > Now take one element out of Y and establish a set A including just this > one element. Take two elements out of Y and establish set B with this > two elements, than take three elements to establish C, and so on. > Since we consider to do this infinitely many times in one step we can > have instantely a new set Z which elements are the sets A, B, C, ... . > So, every element of Z is a set and consists of elements. > You're trying to describe an algorithm that copies all of the elements > of Y, an infinite set, into set Z. Assuming this works, the resulting > set Z is exactly the same set that Y was. There is no elemnt in the set Z which was element in the set Y. What I try to explain is, that if you speak about infinitely many elements of an infinite set, you use the word infinity like a number. So face the truth: In your world exists a maximum element in N: Infinity. On the other side, if you talk about an infinite sequence, you mean a never ending process or extension with no maximum. Two things, one name. How confusing. > If there is such an infinite element in Z, there must be an infinite > number in N - which is impossible. > So there can't be an infinite-valued element in Z. Was there such an > element in Y at the start that got moved from Y into Z? The infinite-valued element in Z is Z. > The Peano axioms specifically allow for the existence of infinite sets. Potentiell infinite sets. > They do not allow for the existence of infinite naturals A natural is a set. > (or, more > precisely, the Peano successor operation only produces finite > naturals). What works for sets is entirely independent for what works > for naturals. Naturals are sets. > The fact that a set has the property of being infinite (i.e., it > contains an infinite number of elements), does not say anything about > the properties of the elements within the set themselves. In same cases yes, in some cases no. Your math world is inconsistent. AS === Subject: Re: Infinity =/= Infinity > What I try to explain is, that if you speak about infinitely many > elements of an infinite set, you use the word infinity like a number. > So face the truth: In your world exists a maximum element in N: > Infinity. No, that is not what infinitely many elements mean. Natural language is often ambiguous and word phrases often have meanings other than the literal reading of each individual word. This can be confusing at first, but once someone tells you what a phrase means, it is ridiculous to argue about it. Does it upset you that pineapples are neither pines or apples? > On the other side, if you talk about an infinite sequence, you mean a > never ending process or extension with no maximum. > Two things, one name. How confusing. Perhaps at first, but it should not take an intelligent person more than short time to figure out the distinctions. In formal settings I do not think the phrase infinitely many is used all that often, and its use can be avoided entirely. However having to phrase everything in such a way that it is impossible for someone to misintrepet, especially when there are people such as yourself who seem intent on misintrepting everything, is tiresome and likely impossible. >> They do not allow for the existence of infinite naturals > A natural is a set. So? You comment would only mean something if you thought that all sets were naturals. Nobody thinks that all sets are naturals. It is possible to think of all naturals as sets, but the two statements are very different. But then again you probably suffer from quantifier dyslexia like most people who have troubles with these concepts. >> (or, more >> precisely, the Peano successor operation only produces finite >> naturals). What works for sets is entirely independent for what works >> for naturals. > Naturals are sets. So? Just because naturals are sets does not mean sets are naturals. Do you really not understand that? Dogs are mammals, but mammals are not dogs. >> The fact that a set has the property of being infinite (i.e., it >> contains an infinite number of elements), does not say anything about >> the properties of the elements within the set themselves. > In same cases yes, in some cases no. > Your math world is inconsistent. You have yet to demonstrate any such inconsistency. You have demonstrated that you are very confused about what words mean, but then again that seems to be common among folks who have problems with infinity. Stephen === Subject: Re: Infinity =/= Infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> Naturals are sets. > So? Just because naturals are sets does not mean sets > are naturals. Do you really not understand that? > Dogs are mammals, but mammals are not dogs. I bet I can find a mammal that _is_ a dog and prove you wrong. Hint: when you are arguing with a person that can't get his concepts straight, you are being less than helpful if you don't get your language straight. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Infinity =/= Infinity Let's have an infinite set Y consisting of elements. Cause ist's a set, > the elements are pairwise different. Now take one element out of Y and establish a set A including just this > one element. Take two elements out of Y and establish set B with this > two elements, than take three elements to establish C, and so on. Since we consider to do this infinitely many times in one step we can > have instantely a new set Z which elements are the sets A, B, C, ... . So, every element of Z is a set and consists of elements. > You're trying to describe an algorithm that copies all of the elements > of Y, an infinite set, into set Z. Assuming this works, the resulting > set Z is exactly the same set that Y was. > There is no elemnt in the set Z which was element in the set Y. > What I try to explain is, that if you speak about infinitely many > elements of an infinite set, you use the word infinity like a number. Informally maybe. Formally, a set with infinitely many elements is one for which a bijection exists between the set and a proper subset. > So face the truth: In your world exists a maximum element in N: > Infinity. No. > On the other side, if you talk about an infinite sequence, you mean a > never ending process or extension with no maximum. No, you mean a function on the set of natural numbers (which is by the way an infinite set). > Two things, one name. How confusing. No, you call an infinite sequence infinite because the set on which it is defined is infinite. Not two things, only one thing. - William Hughes === Subject: Re: Infinity =/= Infinity > Forget all about numbers. > Let's have an infinite set Y consisting of elements. Cause ist's a set, > the elements are pairwise different. > Now take one element out of Y and establish a set A including just this > one element. Take two elements out of Y and establish set B with this > two elements, than take three elements to establish C, and so on. > Since we consider to do this infinitely many times in one step we can > have instantely a new set Z which elements are the sets A, B, C, ... . > So, every element of Z is a set and consists of elements. > card(Z) = infinite = aleph_0 > The question now is: Encloses set Z an element X which consists of > infinitely many elements? > If not - how is it possible that the one set, set Z, reachs infinity in > incrementing but no set X which is an element of Z? Incrementing Z > increments his elements. Z could not go over his elements. > If there is such an infinite element in Z, there must be an infinite > number in N - which is impossible. > Albrecht Storz There are many people which see a set Z and a sequence A, B, C, ... and they argue: Z is infinite cause the sequence A, B, C, ... is infinite, but there is no infinite element in the sequence A, B, C, ... . This is the usual way to missunderstand the difference between an actual and a potential infinity. They also argue, that you can't reach infinity by incrementing numbers or adding elements to finite sets. That must be right. The set Z is a sequence. If we build Z stepwise, what should not make any difference, we have in 1. step a set Z_a which contain one element, the set A. In the 2. step we have Z_b which containes two elements, the sets A and B. In the 3. step we have Z_c which containes A, B, C, and so on. So set Z can be considered as a sequence, the sequence A, B, C, ... . There is no argument why the sequence Z_a, Z_b, Z_c, ... should behave in any other way than the sequence A, B, C, ... . If there is an infinite Z_x in the sequence Z_a, Z_b, Z_c, ... there must be also a infinite X in A, B, C, ... . There is no way out. Albrecht Storz === Subject: Re: Infinity =/= Infinity > There are many people which see a set Z and a sequence A, B, C, ... and > they argue: Z is infinite cause the sequence A, B, C, ... is infinite, As the sequence is a sequence of proper subsets of Z, each of which is a proper superset of the predecessor, the fact that it is infinite does imply Z is infinite. > but there is no infinite element in the sequence A, B, C, ... . What does that have to do with the argument? Z is larger than every set in an infinite sequence of monotonically increasing finite sets. That makes it infinite. > They also argue, that you can't reach infinity by incrementing numbers > or adding elements to finite sets. That must be right. > The set Z is a sequence. No it isn't. > If we build Z stepwise, We can't. - Randy === Subject: Re: Infinity =/= Infinity > Forget all about numbers. > Let's have an infinite set Y consisting of elements. Cause ist's a set, > the elements are pairwise different. > Now take one element out of Y and establish a set A including just this > one element. Take two elements out of Y and establish set B with this > two elements, than take three elements to establish C, and so on. > Since we consider to do this infinitely many times in one step we can > have instantely a new set Z which elements are the sets A, B, C, ... . > So, every element of Z is a set and consists of elements. > card(Z) = infinite = aleph_0 > The question now is: Encloses set Z an element X which consists of > infinitely many elements? > If not - how is it possible that the one set, set Z, reachs infinity in > incrementing but no set X which is an element of Z? Incrementing Z > increments his elements. Z could not go over his elements. > If there is such an infinite element in Z, there must be an infinite > number in N - which is impossible. > Albrecht Storz > There are many people which see a set Z and a sequence A, B, C, ... and > they argue: Z is infinite cause the sequence A, B, C, ... is infinite, > but there is no infinite element in the sequence A, B, C, ... . > This is the usual way to missunderstand the difference between an > actual and a potential infinity. > They also argue, that you can't reach infinity by incrementing numbers > or adding elements to finite sets. That must be right. > The set Z is a sequence. If we build Z stepwise, what should not make > any difference, we have in 1. step a set Z_a which contain one element, > the set A. In the 2. step we have Z_b which containes two elements, the > sets A and B. In the 3. step we have Z_c which containes A, B, C, and > so on. > So set Z can be considered as a sequence, the sequence A, B, C, ... . Correction: So set Z can be considered as a sequence, the sequence Z_a, Z_b, Z_c, ... . > There is no argument why the sequence Z_a, Z_b, Z_c, ... should behave > in any other way than the sequence A, B, C, ... . > If there is an infinite Z_x in the sequence Z_a, Z_b, Z_c, ... there > must be also a infinite X in A, B, C, ... . If there is no infinite element in the sequence Z_a, Z_b, Z_c, ... than Z isn't infinite. Infinity is the maximal element if you want to have actual infinity. Actual infinity leads to logical contradictions. > There is no way out. > Albrecht Storz AS === Subject: Re: Infinity =/= Infinity > Forget all about numbers. Let's have an infinite set Y consisting of elements. Cause ist's a set, > the elements are pairwise different. Now take one element out of Y and establish a set A including just this > one element. Take two elements out of Y and establish set B with this > two elements, than take three elements to establish C, and so on. Since we consider to do this infinitely many times in one step we can > have instantely a new set Z which elements are the sets A, B, C, ... . So, every element of Z is a set and consists of elements. card(Z) = infinite = aleph_0 The question now is: Encloses set Z an element X which consists of > infinitely many elements? If not - how is it possible that the one set, set Z, reachs infinity in > incrementing but no set X which is an element of Z? Incrementing Z > increments his elements. Z could not go over his elements. If there is such an infinite element in Z, there must be an infinite > number in N - which is impossible. > Albrecht Storz > There are many people which see a set Z and a sequence A, B, C, ... and > they argue: Z is infinite cause the sequence A, B, C, ... is infinite, > but there is no infinite element in the sequence A, B, C, ... . > This is the usual way to missunderstand the difference between an > actual and a potential infinity. > They also argue, that you can't reach infinity by incrementing numbers > or adding elements to finite sets. That must be right. > The set Z is a sequence. If we build Z stepwise, what should not make > any difference, we have in 1. step a set Z_a which contain one element, > the set A. In the 2. step we have Z_b which containes two elements, the > sets A and B. In the 3. step we have Z_c which containes A, B, C, and > so on. > So set Z can be considered as a sequence, the sequence A, B, C, ... . > Correction: > So set Z can be considered as a sequence, the sequence Z_a, Z_b, Z_c, > ... . > There is no argument why the sequence Z_a, Z_b, Z_c, ... should behave > in any other way than the sequence A, B, C, ... . > If there is an infinite Z_x in the sequence Z_a, Z_b, Z_c, ... there > must be also a infinite X in A, B, C, ... . > If there is no infinite element in the sequence Z_a, Z_b, Z_c, ... than > Z isn't infinite. Unless the sequence itself is infinite. === Subject: Re: Infinity =/= Infinity > Forget all about numbers. Let's have an infinite set Y consisting of elements. Cause ist's a set, > the elements are pairwise different. Now take one element out of Y and establish a set A including just this > one element. Take two elements out of Y and establish set B with this > two elements, than take three elements to establish C, and so on. Since we consider to do this infinitely many times in one step we can > have instantely a new set Z which elements are the sets A, B, C, ... . So, every element of Z is a set and consists of elements. card(Z) = infinite = aleph_0 The question now is: Encloses set Z an element X which consists of > infinitely many elements? If not - how is it possible that the one set, set Z, reachs infinity in > incrementing but no set X which is an element of Z? Incrementing Z > increments his elements. Z could not go over his elements. If there is such an infinite element in Z, there must be an infinite > number in N - which is impossible. > Albrecht Storz > There are many people which see a set Z and a sequence A, B, C, ... and > they argue: Z is infinite cause the sequence A, B, C, ... is infinite, > but there is no infinite element in the sequence A, B, C, ... . > This is the usual way to missunderstand the difference between an > actual and a potential infinity. > They also argue, that you can't reach infinity by incrementing numbers > or adding elements to finite sets. That must be right. > The set Z is a sequence. If we build Z stepwise, what should not make > any difference, we have in 1. step a set Z_a which contain one element, > the set A. In the 2. step we have Z_b which containes two elements, the > sets A and B. In the 3. step we have Z_c which containes A, B, C, and > so on. The problem is that the fundamental issues are hidden in and so on. Are you attempting to create Z by iteration or not? (Note the fact that Z is a sequence does not mean we need to create it by iteration) > So set Z can be considered as a sequence, the sequence A, B, C, ... . > Correction: > So set Z can be considered as a sequence, the sequence Z_a, Z_b, Z_c, > ... . > There is no argument why the sequence Z_a, Z_b, Z_c, ... should behave > in any other way than the sequence A, B, C, ... . > If there is an infinite Z_x in the sequence Z_a, Z_b, Z_c, ... there > must be also a infinite X in A, B, C, ... . > If there is no infinite element in the sequence Z_a, Z_b, Z_c, ... than > Z isn't infinite. No, there is no reason why an infinite set must contain an infinite element. - William Hughes === Subject: Re: Infinity =/= Infinity > Forget all about numbers. Let's have an infinite set Y consisting of elements. Cause ist's a set, > the elements are pairwise different. Now take one element out of Y and establish a set A including just this > one element. Take two elements out of Y and establish set B with this > two elements, than take three elements to establish C, and so on. Since we consider to do this infinitely many times in one step we can > have instantely a new set Z which elements are the sets A, B, C, ... . So, every element of Z is a set and consists of elements. card(Z) = infinite = aleph_0 The question now is: Encloses set Z an element X which consists of > infinitely many elements? If not - how is it possible that the one set, set Z, reachs infinity in > incrementing but no set X which is an element of Z? Incrementing Z > increments his elements. Z could not go over his elements. If there is such an infinite element in Z, there must be an infinite > number in N - which is impossible. > Albrecht Storz There are many people which see a set Z and a sequence A, B, C, ... and > they argue: Z is infinite cause the sequence A, B, C, ... is infinite, > but there is no infinite element in the sequence A, B, C, ... . > This is the usual way to missunderstand the difference between an > actual and a potential infinity. > They also argue, that you can't reach infinity by incrementing numbers > or adding elements to finite sets. That must be right. > The set Z is a sequence. If we build Z stepwise, what should not make > any difference, we have in 1. step a set Z_a which contain one element, > the set A. In the 2. step we have Z_b which containes two elements, the > sets A and B. In the 3. step we have Z_c which containes A, B, C, and > so on. > The problem is that the fundamental issues are hidden in > and so on. That's right. It is often found like this in math. > Are you attempting to create Z by iteration or > not? (Note the fact that Z is a sequence does not mean we need > to create it by iteration) Who want to worry about it. Could there be a difference if Z is created by iteration or not. How would you create Z else? In which way creates the Peano-Axoims the natural numbers, by iteration or not by iteration? So set Z can be considered as a sequence, the sequence A, B, C, ... . > Correction: > So set Z can be considered as a sequence, the sequence Z_a, Z_b, Z_c, > ... . There is no argument why the sequence Z_a, Z_b, Z_c, ... should behave > in any other way than the sequence A, B, C, ... . If there is an infinite Z_x in the sequence Z_a, Z_b, Z_c, ... there > must be also a infinite X in A, B, C, ... . > If there is no infinite element in the sequence Z_a, Z_b, Z_c, ... than > Z isn't infinite. > No, there is no reason why an infinite set must contain an infinite > element. The reason is as easy as it could be, but hidden under a mountain of desinformation. (PS: I have never said that an infinite set *must* contain an infinite element.) If a sequence like the natural numbers has no maximum element (or the set Z in my example), the cardinality of the set is not defined, because the cardinality could only be the same as the cardinality of the greatest element. If you have a greatest element, that's the cardinality of the whole set. If the greatest element isn't defined, the cardinality of the whole set is not defined. You can call this aspect infinity but than you have to distinguish between an infinite sequence and an infinite set because there is no bijection possible between this two math objects (and this is the raeson why bijecting from the set of natural numbers to their powerset is impossible). You can have sets which are actual infinite but than you must live with infinite numbers. Else you have only potential infinity - a fact which aristoteles had known already. Albrecht Storz === Subject: Re: Infinity =/= Infinity > In which way creates the Peano-Axoims the natural numbers, by iteration > or not by iteration? By fiat. === Subject: Re: Infinity =/= Infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> In which way creates the Peano-Axoims the natural numbers, by >> iteration or not by iteration? > By fiat. Actually, the Peano axioms don't create the natural numbers at all. They just define what makes a set be the set of natural numbers. You can take an arbitrary set and check whether it obeys the Peano axioms. If it does, it is the set of natural numbers. Wherever you got it from. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Infinity =/= Infinity <85ll1wsx8q.fsf@lola.goethe.zz> In which way creates the Peano-Axoims the natural numbers, by >> iteration or not by iteration? > By fiat. > Actually, the Peano axioms don't create the natural numbers at all. > They just define what makes a set be the set of natural numbers. You > can take an arbitrary set and check whether it obeys the Peano axioms. > If it does, it is the set of natural numbers. Wherever you got it > from. > -- It's nothing more than haarspalting what you do. To say creating the natural numbers is just a way of wording. AS === Subject: Re: Infinity =/= Infinity > In which way creates the Peano-Axoims the natural numbers, by >> iteration or not by iteration? By fiat. > Actually, the Peano axioms don't create the natural numbers at all. > They just define what makes a set be the set of natural numbers. You > can take an arbitrary set and check whether it obeys the Peano axioms. > If it does, it is the set of natural numbers. Wherever you got it > from. > -- > It's nothing more than haarspalting what you do. > To say creating the natural numbers is just a way of wording. A good deal of the best of mathematics is little more than creative hairsplitting. === Subject: Re: Infinity =/= Infinity > Forget all about numbers. Let's have an infinite set Y consisting of elements. Cause ist's a set, > the elements are pairwise different. Now take one element out of Y and establish a set A including just this > one element. Take two elements out of Y and establish set B with this > two elements, than take three elements to establish C, and so on. Since we consider to do this infinitely many times in one step we can > have instantely a new set Z which elements are the sets A, B, C, ... . So, every element of Z is a set and consists of elements. card(Z) = infinite = aleph_0 The question now is: Encloses set Z an element X which consists of > infinitely many elements? If not - how is it possible that the one set, set Z, reachs infinity in > incrementing but no set X which is an element of Z? Incrementing Z > increments his elements. Z could not go over his elements. If there is such an infinite element in Z, there must be an infinite > number in N - which is impossible. > Albrecht Storz There are many people which see a set Z and a sequence A, B, C, ... and > they argue: Z is infinite cause the sequence A, B, C, ... is infinite, > but there is no infinite element in the sequence A, B, C, ... . > This is the usual way to missunderstand the difference between an > actual and a potential infinity. > They also argue, that you can't reach infinity by incrementing numbers > or adding elements to finite sets. That must be right. > The set Z is a sequence. If we build Z stepwise, what should not make > any difference, we have in 1. step a set Z_a which contain one element, > the set A. In the 2. step we have Z_b which containes two elements, the > sets A and B. In the 3. step we have Z_c which containes A, B, C, and > so on. > The problem is that the fundamental issues are hidden in > and so on. > That's right. It is often found like this in math. > Are you attempting to create Z by iteration or > not? (Note the fact that Z is a sequence does not mean we need > to create it by iteration) > Who want to worry about it. Could there be a difference if Z is created > by iteration or not. How would you create Z else? [Unless you are very careful, talking about an infinite number of iterations leads to problems. Better to stick with an arbitarily large finite number of iterations.] Create the sequence Z by taking the set of natural numbers and defining a funtion on it. Informally, we create the sequence in a single step. > In which way creates the Peano-Axoims the natural numbers, by iteration > or not by iteration? The Peano Axioms do not create the natural numbers by iteration. The Peano Axioms describe the properties of a single set, the set of natural numbers (which happens to be an infinite set). So set Z can be considered as a sequence, the sequence A, B, C, ... . Correction: So set Z can be considered as a sequence, the sequence Z_a, Z_b, Z_c, > ... . > There is no argument why the sequence Z_a, Z_b, Z_c, ... should behave > in any other way than the sequence A, B, C, ... . If there is an infinite Z_x in the sequence Z_a, Z_b, Z_c, ... there > must be also a infinite X in A, B, C, ... . If there is no infinite element in the sequence Z_a, Z_b, Z_c, ... than > Z isn't infinite. No, there is no reason why an infinite set must contain an infinite > element. > The reason is as easy as it could be, but hidden under a mountain of > desinformation. > (PS: I have never said that an infinite set *must* contain an infinite > element.) > If a sequence like the natural numbers has no maximum element (or the > set Z in my example), the cardinality of the set is not defined, > because the cardinality could only be the same as the cardinality of > the greatest element. No, it is easy to see that the set of natural numbers does not have a largest element but does have a cardinality. > If you have a greatest element, that's the cardinality of the whole > set. > If the greatest element isn't defined, the cardinality of the whole set > is not defined. No, cardinalities are defined in terms of bijections. The fact that no greatest element is defined does not stop a bijection from being defined. (Simple example, map the natural numbers into a proper subset by use of the successor operation.) -William Hughes === Subject: Re: Infinity =/= Infinity > Who want to worry about it. Could there be a difference if Z is created > by iteration or not. How would you create Z else? > [Unless you are very careful, talking about an infinite number > of iterations leads to problems. Better to stick with an arbitarily > large > finite number of iterations.] > Create the sequence Z by taking the > set of natural numbers and defining a funtion on it. Informally, we > create the sequence in a single step. It is totally meaningless in math in which way we create the sequence. (or say: if it is meaningful for math, there must be something wrong). If you create it in one step, you put in infinitely many elements at once (in the case of a infinite set), if you create it stepwise, you put in one element per step infinitely many times. In doing math time, space and matter don't play any part. > In which way creates the Peano-Axoims the natural numbers, by iteration > or not by iteration? > The Peano Axioms do not create the natural numbers by iteration. > The Peano Axioms describe the properties of a single set, the set > of natural numbers (which happens to be an infinite set). You might say: the Peano Axiomes describe the properties of N, or: the Peano Axiomes creates the natural numbers. In math you can't find a difference if the Peano Axiomes just discribes N or creates N. (In spite of this fact, N may exists befor Peano because the Peano Axoimes exists befor Peano.) So set Z can be considered as a sequence, the sequence A, B, C, ... . Correction: So set Z can be considered as a sequence, the sequence Z_a, Z_b, Z_c, > ... . > There is no argument why the sequence Z_a, Z_b, Z_c, ... should behave > in any other way than the sequence A, B, C, ... . If there is an infinite Z_x in the sequence Z_a, Z_b, Z_c, ... there > must be also a infinite X in A, B, C, ... . If there is no infinite element in the sequence Z_a, Z_b, Z_c, ... than > Z isn't infinite. No, there is no reason why an infinite set must contain an infinite > element. > The reason is as easy as it could be, but hidden under a mountain of > desinformation. > (PS: I have never said that an infinite set *must* contain an infinite > element.) > If a sequence like the natural numbers has no maximum element (or the > set Z in my example), the cardinality of the set is not defined, > because the cardinality could only be the same as the cardinality of > the greatest element. > No, it is easy to see that the set of natural numbers does not have > a largest element but does have a cardinality. No. > If you have a greatest element, that's the cardinality of the whole > set. > If the greatest element isn't defined, the cardinality of the whole set > is not defined. > No, cardinalities are defined in terms of bijections. The fact that > no greatest element is defined does not stop a bijection from being > defined. (Simple example, map the natural numbers into a proper > subset by use of the successor operation.) If you put infinitely many elements in a set, you have an infinite set. If you count this elements you will get Infinity. Or you are using two differnt definitions of Infinity. This is your reality cause you are not able to face the truth. My reality is, that there is no actual infinity. But I understand: this reality is too unastonishing for you. So close your eyes and rely on your dreams. Albrecht Storz === Subject: Re: Infinity =/= Infinity > Who want to worry about it. Could there be a difference if Z is created > by iteration or not. How would you create Z else? [Unless you are very careful, talking about an infinite number > of iterations leads to problems. Better to stick with an arbitarily > large > finite number of iterations.] > Create the sequence Z by taking the > set of natural numbers and defining a funtion on it. Informally, we > create the sequence in a single step. > It is totally meaningless in math in which way we create the sequence. > (or say: if it is meaningful for math, there must be something wrong). > If you create it in one step, you put in infinitely many elements at > once (in the case of a infinite set), if you create it stepwise, you > put in one element per step infinitely many times. > In doing math time, space and matter don't play any part. > In which way creates the Peano-Axoims the natural numbers, by iteration > or not by iteration? > The Peano Axioms do not create the natural numbers by iteration. > The Peano Axioms describe the properties of a single set, the set > of natural numbers (which happens to be an infinite set). > You might say: the Peano Axiomes describe the properties of N, or: the > Peano Axiomes creates the natural numbers. In math you can't find a > difference if the Peano Axiomes just discribes N or creates N. (In > spite of this fact, N may exists befor Peano because the Peano > Axoimes exists befor Peano.) Whether or not you think describing N is different from creating N, the Peano Axioms do not create the natural numbers by iteration. > So set Z can be considered as a sequence, the sequence A, B, C, ... . Correction: So set Z can be considered as a sequence, the sequence Z_a, Z_b, Z_c, > ... . > There is no argument why the sequence Z_a, Z_b, Z_c, ... should behave > in any other way than the sequence A, B, C, ... . If there is an infinite Z_x in the sequence Z_a, Z_b, Z_c, ... there > must be also a infinite X in A, B, C, ... . If there is no infinite element in the sequence Z_a, Z_b, Z_c, ... than > Z isn't infinite. No, there is no reason why an infinite set must contain an infinite > element. > The reason is as easy as it could be, but hidden under a mountain of > desinformation. > (PS: I have never said that an infinite set *must* contain an infinite > element.) If a sequence like the natural numbers has no maximum element (or the > set Z in my example), the cardinality of the set is not defined, > because the cardinality could only be the same as the cardinality of > the greatest element. No, it is easy to see that the set of natural numbers does not have > a largest element but does have a cardinality. > No The set of natural numbers does not have a largest element Proof. Assume such an element exists, call it L, Then L+1 is a larger natural number, contradiction. The set of natural number has a cardinality Proof A cardinality is an equivalence class of sets under the relation that two sets are equivalent if there exist a bijection between them. It is not possible for a set not to belong to some equivalence class. . > If you have a greatest element, that's the cardinality of the whole > set. > If the greatest element isn't defined, the cardinality of the whole set > is not defined. No, cardinalities are defined in terms of bijections. The fact that > no greatest element is defined does not stop a bijection from being > defined. (Simple example, map the natural numbers into a proper > subset by use of the successor operation.) > If you put infinitely many elements in a set, you have an infinite set. > If you count this elements you will get Infinity. Or you are using > two differnt definitions of Infinity. I am using cardinalities, which are defined in terms of bijections. Your reply did not mention either. Try again. -William Hughes === Subject: Re: Infinity =/= Infinity >> Are you attempting to create Z by iteration or >> not? (Note the fact that Z is a sequence does not mean we need >> to create it by iteration) > Who want to worry about it. Could there be a difference if Z is created > by iteration or not. How would you create Z else? > In which way creates the Peano-Axoims the natural numbers, by iteration > or not by iteration? The set of natural numbers is not created by iteration. >> No, there is no reason why an infinite set must contain an infinite >> element. > The reason is as easy as it could be, but hidden under a mountain of > desinformation. > (PS: I have never said that an infinite set *must* contain an infinite > element.) > If a sequence like the natural numbers has no maximum element (or the > set Z in my example), the cardinality of the set is not defined, > because the cardinality could only be the same as the cardinality of > the greatest element. Cardinality makes no mention of a largest element. Why not use the actual definitions instead of just making stuff up? Stephen === Subject: Re: Infinity =/= Infinity > Forget all about numbers. > Let's have an infinite set Y consisting of elements. Cause ist's a set, > the elements are pairwise different. > Now take one element out of Y and establish a set A including just this > one element. Take two elements out of Y and establish set B with this > two elements, than take three elements to establish C, and so on. > Since we consider to do this infinitely many times in one step we can > have instantely a new set Z which elements are the sets A, B, C, ... . > So, every element of Z is a set and consists of elements. > card(Z) = infinite = aleph_0 > The question now is: Encloses set Z an element X which consists of > infinitely many elements? If you are asking whether Z contains infinitely many members, yes. If you are asking whether any member of Z contains infinitely many members, then no! > If not - how is it possible that the one set, set Z, reachs infinity in > incrementing but no set X which is an element of Z? Each member of Z, as constructed, contains exactly one more member then the one constructed just previoulsy to it. If there ever were such an infinite member of Z, there would have to be a first or smallest one, as Z is well-ordered by size the of its members, which means that the next smaller one would have to have been finite. Thus you would be requiring a situation in which adding one more member to a finite set produces an infinite set. === Subject: Re: Where do mathematical ideas come from? <7003309.1126211366020.JavaMail.jakarta@nitrogen.mathforum.org So 99% of new math discoveries are plagarisms? > Plagirism = copy verbatim. > Discovery = creatively reframe in your own words to capture the credit > for yourself. > Based on your post, you are not creative enough to plagarize > creatively, so > you, as I am referring only to the creative copiers. Let's try to psychoanalyze you. My theory is that youare too stupid to come up with anything new and so your strategy is to convince yourself that all mathematicians are total moronic cretins who just want to get famous, and thus you still can think of yourself as smarter then them. Yes, it IS painful to realize that you are not the smartest person in the universe, but for all but one person in the universe, this IS true. And believe me, you are NOT that one person. This type of denial strategy doesn't work however. The alternate theory is that you are just a troll trying to get a rise out of people. Jiri === Subject: Re: Where do mathematical ideas come from? strike it rich. I personally know at least one person who is currently writing a calculus textbook. And believe me it is NOT to get rich. It is almost impossible to get a textbook so universally used to get rich. It is apparently tough to get your book used AT ALL. And really if you write a calculus textbook and no college picks it up, then you sell NO copies. Yes some textbook writers can get rich, but I would say very VERY few. There are far easier ways to get rich. For example a dropout of the UCSD phd program started princeton review and made millions (or he started something that became princeton review, I can't remember the story straight). Apparently this is the richest former UCSD mathematics graduate student, and he didn't even past the first year. Jiri === Subject: Re: Where do mathematical ideas come from? > It is conceivable that divorcing Mathematics completely from other > Sciences would result in other Scientists developing their own > Mathematics, which has already happened so often. It is also > conceivable that when Mathematicians try to swallow this half digested > material they do not have a first hand knowledge of what is going on, > so their model may not be adequate. But should it matter? If there is a > need there would be someone else who would improve the model. It would be more interesting to look at examples. There was a lot of mathematics that comes from say physics. For example physicists came up originally with things which we now call distributional derivatives, but in an informal way. It was mathematicians that took those ideas and made them precise and applied these ideas to other parts of mathematics. I think scientists use or create the mathematical tools that they need at the moment. Mathematicians are primarily interested in these tools and their abstractions, rather then their exact application. For example most primality testing routines ASSUME the Riemann hypothesis as true, while we (mathematicians) still don't know if it's true. Even if we knew that the Riemann hypothesis was true far enough for all practical purposes, it still wouldn't satisfy mathematicians, but crytanalysts would not care one bit about it. > On the other hand it always helps to know more and to know that all the > branches of Mathematics originated in efforts to explain some Natural > or Mathematical phenomena or to meet some computational need. So when > we are trying to teach a topic in Mathematics we should bring in the > practical/intellectual/scientific origins of the topic. If this means > including introduction to some physics and some other sciences in the > Math Curriculum, so be it. > Muhammad A common misconception. Mathematics DID NOT originally arise from scientific applications. The ancient greeks were the original pure mathematicians (they also did some applied mathematics, but they clearly didived the two). Most western mathematics before 1700's was based around geometry and number theory. And number theory was only applied most recently. Since people are quoting famous mathematicians, I should too: Before creation God did just pure mathematics. Then He thought it would be a pleasant change to do some applied. -- J. E. Littlewood Jiri === Subject: Re: Where do mathematical ideas come from? > more examples: > a) newton stood on the should of giants and got the > idea to work on > the calculus from liebniz, who also was working on it > at the same time. > b) andrew wiles: he got his fame solving an old > problem somebody else > proposed (fermat) and probably already solved by > somebody else. > c) murray gellman, who discovered quarks. Didn't > the ancient greeks > already propose that matter was made up of > indiviisible pieces? > moral of the story: any new idea you might have has > already been > thought of and probably published by scholars that > came before you. > you just need to know where to dig so that you too > can stand on the > shoulders of giants. So I suppose no one should have gotten credit for West Side Story since it was simply a rip-off of Romeo and Juliet? What discovery do you deem truly unique? Aren't all results novel forms of the same language? Tom === Subject: Re: Where do mathematical ideas come from? > 1) einstein discoverd brownian motion in 1905. But > it up in a dissertation 25 year prior. > Unfortunately, batchelier's > dissertation was filed away and forgotten ... until > einstein secretly > found it and harvested the gold nugget. > 2) einstein discovered general relativity, but > professor riemann > developed all the mathematics. einstein manage to > creatively > re-frame reimann's geometry into a story about > gravity, and staked > claim to all the credit. > 3) einstein discovered the photo-electric effect, > for which he was > awarded the nobel prize. However, some dude in > france discovered the > photoelectric effect 20 years prior, but nobody read > his paper. > 4) Lorentz worked out the theory of transforming > coordinate systems > that left maxwell's equation invariant. Einstein > creatively > re-interpreted Lorentz's system as special > relativity and got all the > credit. I suppose this list is meant to respond to my request for an example of mathematicians who plagiarize (I wish you would post some context for your messages). But it obviously does not come even close to answering my question, so obvious that I refuse to spend time on it, except to the extent that I acknowledge seeing it. Tom === Subject: Re: Where do mathematical ideas come from? > This dialogue seems at cross purposes to me. Isn't > Finite Element Analysis an engineer's tool, after > all? > Well, there's a fairly extensive body of mathematics > used in FEM--approximation theorems in the > appropriate > function spaces and all that. > Which doesn't imply that the engineers who just use > off the shelf packages are familiar with the > mathematical > justification: but then, that's (almost) as true of > calculus. Sure. I would think that, excepting those who only use mathematics to calculate, everyone who does mathematics does so -- at alternate times -- both as a consumer and a creator. When existing tools are not adequate for the problem at hand, new tools must either be found or created. To me, that adequately argues for mathematics as a liberal art, and obviates any demarcation between pure and applied. Tom === Subject: Re: Where do mathematical ideas come from? > This dialogue seems at cross purposes to me. Isn't > Finite Element Analysis an engineer's tool, after > all? > How can one compare it to new theorems, the tools > of the > theoretician? > Do you think there can be no theorems in Finite > Element Analysis? > We use the tools we need to solve the problems that > interest us. There's no difference between pure > and > applied mathematics in that context. > I would translate There's as rather There should > be. > Han de Bruijn I can't think of any demarcation between pure and applied mathematics that is not arbitrary. You yourself just cited the existence (though I can't verify for myself, lacking knowledge) of some theorem structure in FEA, which you deem applied, as opposed to pure, mathematics. Is that theorem structure somehow independent of the theorems of pure mathematics? How? How does the context of applied mathematics differ from the pure? Tom === Subject: Re: Where do mathematical ideas come from? > I can't think of any demarcation between pure and > applied mathematics that is not arbitrary. Agreed. > You yourself > just cited the existence (though I can't verify for > myself, lacking knowledge) of some theorem structure in > FEA, which you deem applied, as opposed to pure, > mathematics. Is that theorem structure somehow > independent of the theorems of pure mathematics? How? As soon as Applied becomes involved with a theorem structure, then that _alone_ is a clear indication that it will _not_ be independent of the theorems of pure mathematics. But more often (than desirable) they become not involved with theorems and proofs. Many of them find that it's quite legitimate to rely on experimental evidence only, by doing numerical experiments without much deep understanding. The FEM which is implemented in the big packages often contain elements that work by sheer luck. I heard the term serependity, but I don't know what it means. > How does the context of applied mathematics differ from > the pure? An example. Once upon a time, there was an engineering office that was involved with the design of grain silos. They calculated the perimeter of these silos as = pi.D , where D is the diameter. And they defined pi as pi = 22/7 .. Now guess what happened, _because_ they were successful. And hence those grain silos became bigger and bigger. Wonder where these increasing gaps came from ... This is a true story. It really happened! Now _that_ is, in some sense, still typical for Applied. That's why I've always said that Applied needs the Pure. But the reverse is also true. Han de Bruijn === Subject: Re: Where do mathematical ideas come from? > I can't think of any demarcation between pure and > applied mathematics that is not arbitrary. > Agreed. > You yourself > just cited the existence (though I can't verify for > myself, lacking knowledge) of some theorem > structure in > FEA, which you deem applied, as opposed to pure, > mathematics. Is that theorem structure somehow > independent of the theorems of pure mathematics? > How? > As soon as Applied becomes involved with a theorem > structure, then > that _alone_ is a clear indication that it will _not_ > be independent > of the theorems of pure mathematics. But more often > (than desirable) > they become not involved with theorems and proofs. > Many of them find > that it's quite legitimate to rely on experimental > evidence only, by > doing numerical experiments without much deep > understanding. The FEM > which is implemented in the big packages often > contain elements that > work by sheer luck. I heard the term serependity, > but I don't know > what it means. > How does the context of applied mathematics differ > from > the pure? > An example. Once upon a time, there was an > engineering office that was > involved with the design of grain silos. They > calculated the perimeter > of these silos as = pi.D , where D is the diameter. > And they defined pi > as pi = 22/7 .. Now guess what happened, _because_ > they were successful. > And hence those grain silos became bigger and bigger. > Wonder where these > increasing gaps came from ... This is a true story. > It really happened! > Now _that_ is, in some sense, still typical for > Applied. That's why I've > always said that Applied needs the Pure. But the > reverse is also true. > Han de Bruijn I hadn't heard that silo story, though it intrigues me. This was in ancient Egypt, perhaps? Tom === Subject: Re: Where do mathematical ideas come from? > Anyway, it was all stolen from the Greeks, who were > perfectly well aware that every material substance > can be analyzed into finitely many elements. That's what we call the naive Finite Element Method. :-) Han de Bruijn === Subject: Proof of convergence for a recursive sequences Underscore sign designates subscript. Thus A_(n+1) means A sub n+1, not A_n + 1. We have two constants k1 =10^-2 and k2=10^-3. Initially we have A_n = 1 and B_n = 0. A_(n+1) = A_n - k1*A_n B_(n+1) = B_n + k1*A_n B_(n+2) = B_(n+1) - k2*B_(n+1) A_(n+2) = A_(n+1) + k2*B_(n+1) Proof that k1*A_(n+R) where R -> infinity will equal k2*B_(n+R) where R goes to infinity. Truly Yours, Simon Dexter === Subject: Re: Proof of convergence for a recursive sequences >Underscore sign designates subscript. Thus A_(n+1) means A sub n+1, not >A_n + 1. We have two constants k1 =10^-2 and k2=10^-3. Initially we >have A_n = 1 and B_n = 0. i.e. for n=0? >A_(n+1) = A_n - k1*A_n >B_(n+1) = B_n + k1*A_n >B_(n+2) = B_(n+1) - k2*B_(n+1) >A_(n+2) = A_(n+1) + k2*B_(n+1) >Proof that k1*A_(n+R) where R -> infinity will equal k2*B_(n+R) where R >goes to infinity. It's not true, but close. If x_j is the vector [ A_{2j} ] [ B_{2j} ] the recurrence can be written x_{j+1} = M x_j where M is the matrix [ 1 - k1 + k1 k2 k2 ] [ .99001 .001 ] [ k1 - k1 k2 1 - k2 ] = [ .00999 .999 ] This has eigenvalues 1 and .98901, a normalized eigenvector for 1 being [ 100 ] u_1 = [ 999 ]/sqrt(1008001) As n -> infinity, M^n will have the limit [ 10000 99900 ] u_1 u_1^T = [ 99900 998001 ]/1008001 [ 1 ] [ 10000/1008001 ] With x_0 = [ 0 ], x_n has the limit [ 99900/1008001 ] So for even n, A_n/B_n approaches 100/999, which is not k2/k1 = 1/10. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === === Subject: GRAVITY IS NOT A FORCE PLANETS ORBIT THE SUN TO CONSERVE TOTAL ENERGY GRAVITATION IS NOT A FORCE BUT AN ILLUSION Copyright 1984-2005 Allen C. Goodrich A planet or any mass such as the earth orbits the sun simply because it would require the gain or loss of a tremendous amount of energy to make it travel in any other orbit or path.This is the only path where its kinetic and potential energies,relative to the rest of the universe, are equal in magnitude, and their sum is a constant. But,why do we seem to be attracted to the earth by a force of gravity? SUMMARY OF PAST HISTORY: The precise measurements of planetary motion by Tycho Brahe (1546-1601) and observations by Galileo Galilei (1564-1642) were plotted by Johann Kepler (1571-1630 ) resulting in Kepler's Three laws: 1. The planets move about the sun in elliptical orbits with the sun as one focus of the ellipse. 2. The straight line joining the sun and a given planet sweeps out equal areas in equal intervals of time. 3. The square of the period of revolution of the planet about the sun is proportional to the cube of the mean distance from the sun. t^2 = K L^3 Sir Isaac Newton (1642-1721 ) concluded that it was a force F = mL/t^2 = k m_1 x m_2 /L^2 that caused the orbital motion. Allen C. Goodrich defined the cause as a conservation of total energy. The concentration of the Kinetic Energy of mass increases as the Potential Energy of the universe decreases with the expansion of the universe at constant total energy. Planets orbit the sun in a state of equiliurium,where no change to total energy occurs. At Equilibrium the sum of kinetic and potential energies is a constant. A positive change of kinetic energy equals a negative change of potential energy. + delta m (2 pi L)^2/t^2 = - delta G (M-m)m / L . or Delta e (2 pi L)^2/t^2 = - Delta K e^2 / 4 pi E_o L. if a charge is present. The mass of the human body, on the earrth's surface, is not in an equilibrium orbit. If a force ,such as the surface of the earth , was not present, the body would not stay where it is. IT TRIES TO MOVE TO AN EQUILIBRIUM ORBIT.( No change of total energy) This force is what is felt to rqual gravitational force. A gravitational force is not needed in a state of orbital equilibrium. Galileo demonstrated the effect of gravitational force. Newton assumed that a gravitational force between all masses pulled them together. Was this a correct assumption? Einstein and many other scientists felt that there must be more to gravitation than an attraction at a distance. Action at a distance was considered to be impossible in the absence of a transfer of energy at the speed of light. A change of kinetic energy is not always the result of a force. In an equilibrium system at constant total energy, kinetic energy can increase as potential energy decreases, with the total energy remaining constant.. Hubble then showed that the distant Galaxies were moving away from the earth and that the universe was expanding in all directions. If this is true , What else must be true? 1. The potential energy of the rest of the universe must be decreasing relative to the mass of the earth. It has long been assumed that the first law of thermodynamics, which says that the total energy of the universe is a constant, was a fact of nature. If this is true what then? 2. The kinetic energy of the universe must be increasing at the same rate that the potential energy is decreasing as the universe expands. How is this possible? Masses must be accelerating, because, kinetic energy is the result of an acceleration. 3. Orbital motion could then be the result of the expansion of the universe. The Gravitational illusion could be the result. Based on the first law of thermodynamics The total mass energy of the universe is a constant. ((total kinetic (mass) energy plus total potential energy is a constant)). m is any mass say that of the earth. Planets, moons, and electrons are normally in equilibrium orbits where the total energy is constant. m(2 pi L)^2/t^2 + G(M-m)m/L+ X e(2 pi L)^2/t^2 + Z e^2/4 pi E_o L = a constant. In the absence of a charge, from this equation the equation Delta m (2 pi L)^2 / t^2 = - Delta G (M-m)m/L follows mathematically. The earth orbit is a result of an energy equilibrium, ( the absence of a change of total energy ) and not the result of a force of gravity between masses. Force of gravity is the resulting illusion assumed by Newton to be a force. If a planet (say earth) moved away from the sun its potential energy would decrease as L increased. Its kinetic energy would decrease because it is no longer accelerating toward the sun in orbital motion. Total energy would have to decrease. A very great change of total energy would have to take place. POTENTIAL ENERGY = G(M-m)m/L KINETIC ENERGY = m(2 pi L)^2/t^2 m(2 pi L)^2/t^2 + G(M-m)m/L = A constant = M G= Gravitational constant; M = total energy of the universe (or effective universe) ; m = mass in question. t = time ; L = radial distance. No mechanism exists for this to occur rapidly. So it could not happen. The magnitudes of kinetic and potential energies of planets and moons travelling in orbital motion are nearly equal and any increase or decrease of orbital distance L results in an equal change in magnitude of both.This is the only value of L where no change of total energy will occur if the value of L changes. At any other distance L, an increase of kinetic energy will be at a different rate than potential energy decreases. Orbital motion conserves total energy. Force of gravity isn't needed to explain orbital motion or any other motion at a distance. GRAVITY MECHANICS AND RESEARCH ON ASTRONOMICAL OCEAN TIDES Copyright 1984 to 2002 Allen C. Goodrich An examination of United States Coast and Geodetic Survey Tidal Data, which was gathered by extensive measurements over long periods of time,was compared with astronomical data showing the phases of the moon at corresponding times for many years. This correlation of the two sets of data revealed a very interesting fact, in a manner that had never before been mentioned in the literature. It is invariably and exactly the lowest tide that exists directly under the full and new moons at deep ocean ports. TABULATED co-op.nos.noaa.gov and space.jpl.nasa.gov DATA: OCEAN TIDES AND PHASES OF THE MOON AT DEEP OCEAN PORT- MYRTLE BEACH LOWEST TIDE (YEARS 1992 AND 1993) 1992 FULL MOON---1992 NEW MOON (at moons highest point in the sky) DATE---TIME(std)-DATE---TIME(std) Mar.18--12:00Mid-Mar.3---12:00Noon Apr.17--12:00Mid-Apr.2---12:00Noon May.17--12:00Mid-May.2---12:00Noon Jun.15--12:00Mid-Jun.29--12:00Noon July.13-12:00Mid-July.29-12:00Noon Aug.12--12:00Mid-Aug.27--12:00Noon Sept.11-12:00Mid-Sept.26-12:00Noon Oct.11--12:00Mid-Oct.26--12:00Noon Nov.10--12:00Mid-Mov.25--12:00noon Dec.10--12:00Mid-Dec.25--12:00noon 1993 FULL MOON---1993 NEW MOON (at moons highest point in the sky) DATE---TIME(sdt)-DATE---TIME(sdt) Jan.8--12:00Mid--Jan.24-12:00Noon Feb.6--12:00Mid--Feb.21-12:00Noon Mar.8--12:00Mid--Mar.23-12:00Noon Apr.6--12:00Mid--Apr.21-12:00Noon May.6--12:00Mid--May.20-12:00Noon Jun.4--12:00Mid--Jun.19-12:00Noon July.3-12:00Mid--Juy.18-12:00Noon Aug.2--12:00Mid--Aug.17-12:00Noon Sep.1--12:00Mid--Sep.16-12:00Noon Sep.30-12:00MId--Oct.15-12:00Noon Oct.30-12:00Mid--Nov.14-12:00Noon Nov.29-12:00Mid--Dec.13-12:00Noon Dec.28-12:00Mid--Jan.12-12:00Noon This was a very interesting discovery because current physics,based on the gravitational theory, discussed in the following U.S.Gov. documents: PREDICT THE OCEAN TIDES http://co-ops.nos.noaa.gov/restles1.html SEE PHASES OF THE MOON FROM EARTH http://space.jpl.nasa.gov/ ,would lead one to believe that,except for many possible reasons, the highest tides tend to be under the full and new moons. The dictionary and encyclopedia as well as physics texts predict this with pictures of the earth and oceans bulging on the side facing the full moon. Of course it never happens as the gravitational theory predicts, and many reasons are given for the discrepancies. CONCLUSION: No discrepancies were found in the occurence of exactly the lowest tide directly under the full and new moons, at deep ocean ports. A lowest tide also occurs on the earth's ocean directly opposite to the new and full moons. SIGNIFICANCE: One must admit that this is beyond question one of the most important discoveries of modern physics research. It indicates that a change must be made in the theory of gravitation. One can no longer assume that a force between the moon and the water of the earth's oceans, is causing the ocean tides. The force of gravity must be an illusion caused by some other, more basic, reason. What would this be? If the total energy ( kinetic and potential ) of the universe is assumed to be a constant,from this fundamental equation, many interesting things follow. If the rest of the universe is expanding ( potential energy decreasing) relative to masses, the masses must be shrinking ( increasing in kinetic energy ) (gravitation) relative to the rest of the universe. THE FIRST LAW OF MOTION-(GOODRICH) Copyright 1984 to 2002 ALLEN C. GOODRICH A body (m) continues in a state of rest (equilibrium) or motion in a straight or curved line (equilibrium) as long as no change occurs in its total (kinetic and potential) energy, relative to the rest of the effective universe (M-m), Delta m(2 pi L)^2/t^2 = - Delta K(M-m)m/L equilibrium = no change in the total energy relative to the rest of the effective universe (M-m). ^ = to the power of. Orbital motion complies with this equation. This equation is derived from the fundamental equation of the universe which states that the total energy of the universe is a constant. The sum of kinetic and potential energies is a constant. m(2 pi L)^2/t^2 + K(M-m)m/L = A constant. INERTIA AND MOMENTUM are the properties of a mass that evidence its reluctance to change its total energy, or it is its need to maintain a constant total energy. If it could more easily obtain or lose energy, it would have less inertia or momentum. SEE THE UNIVERSE- A GRAND UNIFIED THEORY OF MASS ENERGY SPECTRUM OF THE BUFFALO ASTRONOMICAL ASSOCIATION INC. NOV.1996 TO FEB.1997 :( CLICK BLACK AND BLUE PAGES BELOW ) http://ourworld.cs.com/gravitymechanic2/myhomepage/business.html http://ourworld.cs.com/gravitymechanic2/myhomepage/profile.html TIDES AND GRAVITY MECHANICS http://ourworld.cs.com/gravitymechanic2/myhomepage/resume.html A new theory of gravitation is given, which predicted, stimulated the above research,and is consistent with, the new findings. The universe has been found to be expanding at an accelerating rate as predicted in 1984 by this new theory. ELECTROMAGNETIC ,PHOTON AND CHARGE EFFECTS. ARE DEFINED IN THE FOLLOWING BOOK.-- THE UNIVERSE( ISBN 0-9644267) library of congress catalog no. 94-90554:--Allen C. Goodrich Copyright 1984 to 2005 Allen C. Goodrich FORCE OF GRAVITATION DOES NOT EXIST. If One calculates the kinetic and potential energies of the planets relative to the rest of the effective universe, using the formulas kinetic energy = m(2 pi L )^2/t^2 and potential energy = -G(M-m) m/L, M is the gm mass of the sun and all planets; m ,L,and t are the gm mass, mean radial cm. distance, and orbital time in sec, of one of the planets. ( THIS IS THE ONLY CORRECT METHOD, it explains the T.R.Young-two slit interference pattern which involves the rest of the universe ). One will find that they are of nearly equal magnitude but opposite in sign. One will also find that their sum is a constant, the equilibrium energy for the particular planet.This is the energy that remains constant as the universe expands. its potintial energy continually decreasing and its kinetic energy continually increasing. Only at the orbital distance will a small change of kinetic energy equal an opposite change of potential energy.This is the total energy that requires no force , with its necessary acceleration and change of total energy, to maintain it as a constant.No force of gravity is necessary to explain the motion of the planets in the expanding universe. The planets motion around the center of the rest of the universe at the specific distance L is the equilibrium condition for constant total energy of the orbiting planet in the expanding universe. THE SOLAR SAIL Copyright 1984 to 2005 Allen C. Goodrich The Solar Sail, which is being tested by Russia and the United States, for possible propulsion in interstellar space travel, is additional evidence that no change of potential energy to kinetic energy of the photon takes place unless the potential energy is absorbed .The photon does not have mass ( kinetic energy). A change of direction of the photon's potential energy can occur at the reflective surface but no potential to kinetic energy change takes place there. A change of potential to kinetic energy takes place at the black absorption surface.which has the correct frequency response as well as direction and density (time ) in the expanding universe.This is evidence that the photon is potential not kinetic energy.The light photon does not have mass or kinetic energy.until the photon is absorbed by a mass of the correct frequency response as well as direction and density (time ), no potential to kinetic energy change can take place.in the expanding universe, in the absence of a mass.. THE VELOCITY OF LIGHT IS AN ILLUSION Copyright 1984 to 2005 Allen C. Goodrich A negative kinetic energy change of a mass, is a positive potential energy change of the rest of the effective universe relative to a mass of the proper frequency, direction ,distance L and time change t (density), in the expanding universe. This is consistent with the first law of thermodynamics, whch conserves total energy.. The L/t is currently falsely assumed to be a velocity of light. This explains the T.R. Young two slit interference pattern. change of the entire universe, that can become a positive kinetic energy change of a mass such as the electron if the frequency, direction, distance L , and time change t (density) are correct.. === Subject: Quadratic forms Consider a quadratic form (on R^3) 'xAx and suppose A is diagonal, like a 0 0 0 b 0 0 0 c . This is my question: if we want to express 'xAx in the form 'xSx, where S is a matrix that can have only 1,-1 or 0 as diagonal entries (which existence is guaranteed by Sylvester's diagonalization theorem), we act with a projective transformation? P.S. If I had to answer to myself, I would say: yes (but obviously I'm not sure...), because for example we can put X=Mx (that should be a *projective X1 X2 X3 A 0 0 a 0 0 A 0 0 X1 0 B 0 0 b 0 0 B 0 X2 0 0 C 0 0 c 0 0 c X3 === Subject: Re: Relationship between mathematics and music <85slw615s8.fsf@lola.goethe.zz> <5--dndJzTrNBS7TeRVn-pg@whidbeytel.com> Music is no more mathematical than any other human activity. No more >> mathematical than gambling, origami, fishing, pole-vaulting or farting. >> Certainly less mathematical than manufacturing, marketing, linguistics, >> engineering, physics, chemistry, statistics, or sailing. > Sailing? > Yes, of course sailing. For example, the America's Cup competition is as > much an engineering (i.e. mathematical) contest as it is a sailing race. [Snip description of sailboat design] Again you seem unable or unwilling to distinguish between composition and performance. In every human endeavor those are two different things. There's lots of science that goes into training olympic runners, but olympic runners are not, as a rule, scientists or engineers, so I wouldn't make a claim that running is heavily scientific, only that training of runners is heavily scientific. You haven't made a case here that sailing is a mathematical endeavor, only that design of sailboats is. Except here: > There are also sailing strategies that are studied in mathematical terms to > help give the skipper gain every imaginable advantage. Even weather > prediction, which is hugely mathematical, plays a role in selecting the > skipper's strategy in advance of the race. Oh yeah. and don't forget knot > theory (just kidding)! > OK. Does even the most hideously baroque What does hideously baroque mean or most baroque mean? > music composition get into that much mathematics? I believe I've already explained how the COMPOSITION of multi-voice fugues does. It's the same kind of mathematics as construction of magic squares or latin squares. Look them up. I don't think that PERFORMANCE of baroque compositions requires mathematical skills. I do believe that performance of jazz requires a theoretical understanding of the structure of a piece that classical does not. I remember a story on the radio, a famous jazz musician who described his comeuppance early on upon joining a well-known band. He did pretty well with his audition, he thought. Then the bandleader told him to do it again in D. He'd given no thought at all to the essentially mathematical skills needed for that. You could transpose a piece of music even if you had no musical performance abilities at all. You could take the notation in one key and convert it to another key even if you had no ability to sight read, to hear in your mind what sounds those marks corresponded to. It's a purely mathematical (well, more arithmetical) skill. - Randy === Subject: Re: Relationship between mathematics and music > relationship between music and math? Particularly, I am interested in > what mathematically makes a musical composition sound good. I recently > bought The Idiots Guide to Music Theory and so far it's very good. Here's my review of three such books (plus a bit): http://imaginatorium.org/books/mathmus.htm Brian Chandler http://imaginatorium.org === Subject: Re: Relationship between mathematics and music > relationship between music and math? Particularly, I am interested in > what mathematically makes a musical composition sound good. I recently > bought The Idiots Guide to Music Theory and so far it's very good. > Here's my review of three such books (plus a bit): > http://imaginatorium.org/books/mathmus.htm Interesting. The Music of the Spheres review gets into the mathematics of the 12-tone scale. This reminds me that I've heard some composers have actually done serious work with scales other than 12-tone that make mathematical sense (and aesthetic sense). I don't know very much about this and I've never heard any of this music. - Randy === Subject: Re: Relationship between mathematics and music > Interesting. The Music of the Spheres review gets into the > mathematics of the 12-tone scale. This reminds me that I've > heard some composers have actually done serious work with > scales other than 12-tone that make mathematical sense > (and aesthetic sense). I don't know very much about this > and I've never heard any of this music. Several non-12-note scales have been in use in Indian music for a hell of a long time now. === Subject: Re: Relationship between mathematics and music mathematics of the 12-tone scale. This reminds me that I've > heard some composers have actually done serious work with > scales other than 12-tone that make mathematical sense > (and aesthetic sense). I don't know very much about this > and I've never heard any of this music. > Several non-12-note scales have been in use in Indian music for a hell of a > long time now. I had in mind the theorists working on the harmonic theory of such scales. Something like this site: http://tonalsoft.com/enc/encyclopedia.aspx The magic word seems to be microtonal. This looks like a better site: http://www.microtonal.co.uk/index.html This page mentions 19, 31, and 43-tone divisions. For mathematical reasons, I gather these numbers aren't arbitrary. I think that 43 was the number I heard when I first heard of this stuff. http://infohost.nmt.edu/~jstarret/microtone.html Ah, one more bit of info. The composer that was mentioned to me in this connection: Harry Partch. http://www.corporeal.com/cm_main.html - Randy === Subject: Re: Relationship between mathematics and music > relationship between music and math? Particularly, I am interested in > what mathematically makes a musical composition sound good. I recently > bought The Idiots Guide to Music Theory and so far it's very good. > Craig Most replies so far are more about harmony than about compositions. that could analyse existing compositions and produce new ones in the same style. (Btw, Cope has a homepage.) This typically works very well with all music in which 'musical idiom' plays a key role, rather than the genius of the composer. E.g. the computer's imitations of Chopin Mazurkas or ragtime music are definitely recognizable as music in the same style. With Debussy or Scriabin, the imitations are euh... much further away from the original. What Cope's work showed is: how to do mathematical representation of musical 'idiom' is reasonably clear, now. But apart from musical idiom, there's something more that makes music worthwhile. But we still can't put our finger on what exactly. (And afaik, no improvements on Cope's work exist, yet.) -- Herman Jurjus === Subject: Re: Relationship between mathematics and music <432a79cb$0$30355$ba620dc5@text.nova.planet.nl> See http://arxiv.org/abs/cs/0303025 for a clever way to classify music (jazz, rock, classical) without knowing a thing about music, using algorithmic information theory. I wonder if their technique would work to separate great music from mediocre/bad music. That way we could predict whether a song is good without listening to it!! Craig === Subject: Re: Relationship between mathematics and music !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > See http://arxiv.org/abs/cs/0303025 for a clever way to classify > music (jazz, rock, classical) without knowing a thing about music, > using algorithmic information theory. I wonder if their technique > would work to separate great music from mediocre/bad music. > That way we could predict whether a song is good without listening > to it!! I don't think so: there are artforms that more or less _identify_ themselves over compression. Roy Lichtenstein, for example, has made reduction to the information content of coarse color prints an art form, and Pablo Picasso has envisioned JPEG compression artifacts before the age of computing. You could not usefully place their works in a scale based on information content. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: infite primes mod x The problem is: === Subject: Re: infite primes mod x days. My association with the Department is that of an alumnus. >The problem is: An argument similar to the classical argument that shows there are infinitely many primes will work. First, note that if a number is divisible only by primes which are of the form 1 mod 4, then the number must itself be of the form 1 mod 4. Now, let {p_1,...,p_n} be any finite list of primes which are of the form 3 mod 4. We only need to show that there is at least one prime of the form 3 mod 4 which is not in the list. Consider N = (p_1*...*p_n)^3 + 2. -- === Subject: New version of 3D-XplorMath visualization program === Subject: New version (10.4.1) of mathematical visualization Software, 3D-XplorMath This announcement will primarily be of interest to Macintosh users, however note the announcement below of an upcoming Java version. 3D-XplorMath version 10.4.1 is a freeware mathematical visualization tool that is a much improved version of the well-known program 3D-Filmstrip. NOTE: The 3D-XplorMath webserver has moved. The new server, located at the University of California, Irvine Mathematics Department, is vmm.math.uci.edu. The new 3D-XplorMath homepage is at: http://vmm.math.uci.edu/3D-XplorMath/ and the program together with full documentation can be downloaded from there. (There is a Gallery of 3D-XplorMath produced visualizations and animations at: IMPORTANT NEWS!: Professor David Eck of Hobart and William Smith Colleges is creating a cross-platform Java program called VMM (for the Virtual Mathematical Museum) that will eventually include all of the essential functionality of 3D-XplorMath. Work on this is progressing very well, and we hope to have a public release of an early version by Spring of 2006 if not earlier. Changes to 3D-XplorMath between versions 10.4 and 10.4.1 We have made a considerable number of small bug fixes and improvements to the user interface. In particular, there are numerous speed-ups included in this version. The Plane Curve Category: Now most planar curves come with a decoration that explains how the curve is defined. The latest addition is a mechanical construction of the Lemniscate. We have also made the various decorations perform in a more uniform fashion. The Space Curve Category: We have added Curves of Constant Curvature to the exhibits (with many closed ones in the default morph) and curves of constant torsion (again closed ones in the default morph). We had not seen closed space curves of constant curvature treated before and found them interesting to look at. Although computed from their Frenet differential equation, all of the entries for explicitly parametrized curves in the Action Menu remainavailable for this new exhibit, and the same holds for Curves of Constant Torsion. See the ATOs for further interesting details. The Surface Category: In the minimal surface subcategory there is a new Action Menu item: Show Associated Grids. It shows on the left the grid on which we perform the numerical integration, in other words, this grid defines which parameter lines appear on the surface. On the right we either show the Gauss image of the integration grid (left), or, for surfacesof genus > 0, we show the image grid under the complex third coordinate function. The Fractals & Chaos Category: Instead of showing only still pictures of the Henon attractor and of the Feigenbaum tree we have added some dynamics so that one can now see how the final pictures evolve from early approximations. The use of colors shows clearly where a mixing behaviour occurs. Also, in the Henon Attractor, if you hold down Command, you can drag out a zoom rectangle on the screen (i.e., the window will zoom to a magnified view of the part of the attractor included in that rectangle). Important improvements have been made to the Julia Set animations. Program Description: 3D-XplorMath (formerly 3D-Filmstrip) is a highly interactive museum for exploring the visual aspects of the beautiful universe of mathematical objects and processes. It has been under continual development for more than ten years by an international team of mathematical researchers who originally perfected it for their own use in teaching and research, but have recently been working to make it easy and enjoyable to use by anyone with mathematical curiosity and an appreciation for the visual and logical beauty of math. This museum contains literally hundreds of well-known (and some not so well-known) mathematical objects, arranged logically into numerous galleries, referred to as Categories. These include: Surfaces, Planar Curves, Space Curves, Polyhedra, Conformal Maps, Dynamical Systems, Waves, and (the latest) Fractals & Chaos. The 3D in its name refers to the fact that 3D objects can be viewed in strikingly realistic stereo. 3D-XplorMath differs from programs such as Mathematica, Maple, and Matlab that, while providing visualization back-ends for viewing objects, require the user to first program the object and visualization. 3D-XplorMath emphasizes ease of use and does not require the user to have a pre-existing knowledge of the mathematical definition of an object in order to see it. Every mathematical object in its massive collection is not only pre-programmed, but also has carefully chosen default parameters and associated animations. Merely selecting a gallery object by its name from a menu presents an excellent initial view of the object. The user may then optionally use simple dialogs, controls, and menu choices to customize and animate the default view, perhaps after first learning about its background by choosing About This Object from the Documentation menu. Users can also create and animate new objects on their own by entering simple algebraic formulas into dialogs. All objects including user defined objects can be saved in several graphic formats, and animations can be saved as Quicktime movies. === Subject: quiz question for graduate students of math In the past I have on occasion given quiz questions to graduate students and today I quizzed the students of Harvard, Princeton, Yale on a physics question. So let me quiz the MidWest for there are alot of bright and sharp graduate students in the Midwest. I want to stay away from the South because I believe the heat and hot climate is not conducive to good minds and that the best of human civilization was produced in the cooler climates. And why I would have thought that space travel in the USA should have found its home base not in Florida or Texas but in the northern states where it is cooler and less prone to hurricanes. Another example of where political pork barrel renders a less than optimal foundation for space travel. Question to the fine students of the Midwest graduate schools: Recently I bought a number of 2 by 6 lumber 12' long and will cut about 8 off the end of each board. I will construct a bench pattern of board spaced by those pieces of cutoff. Something like this |s|s| | | | | | | | | | |s|s| where the lines are boards and s signifies a spacer. Many have seen this pattern in benches where 2 by 4 boards are spaced apart by spacers and so there is a grill like pattern and if you have a coin it can fall between the boards of the bench. So the question is, as I saw each board of an 8 piece off the end and use it as a spacer that for 3 boards I need 4 spacers as the diagram above shows, so I have to come up with one more spacer if I had only 3 boards in all. But if I had say 12 boards that would mean I have 12 spacers that I cut, so how many boards can I assemble with 12 spacers? I believe it is 7 boards. So what is the general formula for any n boards? Pity that this problem is easier than the physics problem. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: quiz question for graduate students of math I think that any carpenter would tell you; for any n boards you need exactly 1 and only 1 spacer because you can reuse the silly thing after you have the seat board fastened (or even marked well) to the cross member at each support structure in the bench. 2 Spacers would allow your work to go faster, and if you were making lots of benchs and had some apprentice carpenters you might need 2 spacers for each one to obtain maximum efficiancy. A more interesting consideration might be the maximum number of spacers any given carpenter should use in order to produce the most benchs in the shortest amount of time assuming quality control will kick back crappy benchs. As for your remarks on the south many a great mind was born in the warmer climes. But aside from individuals great cultures have come from warmer climes as well. Consider the Myans, Babylonians, Greeks, Egyptions just to name a few all of these cultures made huge contributions to science and mathamatics. Granted many of the advances that these peoples discovered are trivial now, I think Issac Newton said it best and I paraphrase... We are here now because we stand on the shoulders of giants. I wont even get into the physics of why the space program is located in the deep south of the US. === Subject: Re: quiz question for graduate students of math > As for your remarks on the south many a great mind was born in the > warmer climes. But aside from individuals great cultures have come > from warmer climes as well. Consider the Myans, Babylonians, Greeks, > Egyptions just to name a few all of these cultures made huge > contributions to science and mathamatics. One more case in point... The Viking age flourished suddenly at a time when there was a temporary global warming phase underway. Many historians think that it was this rather slight increase in temperature that kicked the agriculture and population growth of Norway and nearby areas, resulting in the emergence of a new civilization. The Viking age resulted in great developments in sea travel, weapons technology, and poetry. === Subject: Re: quiz question for graduate students of math that is an interesting remark I was going to mention that aside from cultures that existed in warm climates no other cultures had really made a lot of progess on anything except the Vikiings. I mean look at the dark ages in europe. These people only got motivated back into thinking after they came into contact with middle eastern scholars and traders from the far east or atleast middle easterners that met up with traders from the far east. It is clear that not only does the original poster have a serious lack of knowledge regarding the mathamatics of benchs and the physics of space launches but he has a serious lack of understanding of history as well. I do think however his corny theory of the atom as the universe website is a riot. I bet he would argue that was his idea. And it came to him in the winter in North Dakota. Ofcourse I would be willing to bet dollars to pennies that he was in a nice heated house at the time. === Subject: Re: quiz question for graduate students of math > In the past I have on occasion given quiz questions to graduate > students and today I quizzed the students of Harvard, Princeton, Yale > on a physics question. So let me quiz the MidWest for there are alot of > bright and sharp graduate students in the Midwest. I want to stay away > from the South because I believe the heat and hot climate is not > conducive to good minds and that the best of human civilization was > produced in the cooler climates. Counter example: The brilliant Paul Morphy was born in New Orleans, Louisiana - one of the very greatest chess masters of all time. Counter Example: The genius Louis Armstrong also born in New Orleans - created a new art form. > And why I would have thought that > space travel in the USA should have found its home base not in Florida > or Texas but in the northern states where it is cooler and less prone > to hurricanes. Another example of where political pork barrel renders a > less than optimal foundation for space travel. Nope. It is less expensive to launch closer to the equator... But to know that, you would have to know some grade 12 physics. [stupid quiz snippet] === Subject: Re: quiz question for graduate students of math <9K6dneiJDJgfmrbeRVn-3Q@whidbeytel.com> Counter example: The brilliant Paul Morphy was born in New Orleans, Louisiana - one of the very greatest chess masters of all time. Counter Example: The genius Louis Armstrong also born in New Orleans - created a new art form. It would be nice if you knew what counterexamples were. A counterexample to my statements would be to show us where science and technology flourished in the heat of tropical or subtropical climates. It never did. Human civilization is concentrated in the Temperate climate zones and that is because science and technology flows best out of a cool mind. Heat conditions the mind to not be active. Give me an example of a famous scientist who lived all of his/her life in the tropics and subtropics and discovered something of great importance whilst in that hot climate. The only person I can think of is Dirac who retired to Florida but whilst he was in Florida, Dirac never had any important new idea. I am not saying that the heat will prohibit the discovery of any great new science idea, but that someone up north in a cooler climate will beat anyone in the south to that great new idea, simply because heat and hotness are not conducive to great ideas in science. If you think you have a good mind for physics or science and you have a choice of living either in a hot and dirty environment such as California of Stanford and CalTech or living in a cool environment that is cleaner in air such as say Minnesota or Wisconsin or Iowa then it is more likely that you discover a great new idea in physics by living in Minnesota or Iowa or South Dakota rather than hot and dirty California. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: quiz question for graduate students of math <9K6dneiJDJgfmrbeRVn-3Q@whidbeytel.com Counter example: The brilliant Paul Morphy was born in New Orleans, > Louisiana - one of the very greatest chess masters of all time. > Counter Example: The genius Louis Armstrong also born in New Orleans - > created a new art form. > It would be nice if you knew what counterexamples were. A > counterexample to my statements would be to show us where science and > technology flourished in the heat of tropical or subtropical climates. > It never did. Human civilization is concentrated in the Temperate > climate zones and that is because science and technology flows best out > of a cool mind. Heat conditions the mind to not be active. > Give me an example of a famous scientist who lived all of his/her life > in the tropics and subtropics and discovered something of great > importance whilst in that hot climate. Archimedes (lived in Sicily). Now why didn't you think of him? Euclid (lived in Egypt). -- J. H. Palmieri University of Washington === Subject: Re: quiz question for graduate students of math <9K6dneiJDJgfmrbeRVn-3Q@whidbeytel.com> How close to the Equator do the Russians and Europeans launch their space rockets. You fool. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: quiz question for graduate students of math > How close to the Equator do the Russians and Europeans launch their > space rockets. You fool. > Archimedes Plutonium > www.iw.net/~a_plutonium > whole entire Universe is just one big atom > where dots of the electron-dot-cloud are galaxies Wow.. you are stupid. The European Space Agency launches are done from French Guiana, South America... Much closer to the equator than Florida, you fool. Russian rockets will soon be launched from the French Guiana starting in in 2008, as a result of an agreement signed recently. Up till now, Russia usually launches rockets from Baikonur, Kazakhstan, wjich is about as close to the equator as you can get. The Russians paid a lot of extra costs to keep their launches inside the old USSR territory. I guess it was to keep it secret and under control. Idiot! === Subject: Re: quiz question for graduate students of math <9K6dneiJDJgfmrbeRVn-3Q@whidbeytel.com> Kazakhstan is about 45 degrees north latitude. Minneapolis and North and South Dakota are about 45 degrees north latitude. The latitude never bothered the Russians. So it would have been a far better site location for the USA space program to have been located in South Dakota or North Dakota or Minnesota where there is no danger of hurricanes and where cooler minds can work in a cooler environment. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: quiz question for graduate students of math > Kazakhstan is about 45 degrees north latitude. Minneapolis and North > and South Dakota are about 45 degrees north latitude. The latitude > never bothered the Russians. So it would have been a far better site > location for the USA space program to have been located in South Dakota > or North Dakota or Minnesota where there is no danger of hurricanes and > where cooler minds can work in a cooler environment. If Kazakhstan was so great, then why did the Russians pay the European Space Agency for access to the South American launch site for future launches? Just for the Pina coladas? Hurricanes can be a risk, but NASA uses caution, and there are several days warning notice to put away their toys if required. In the Northern US, other problems are more serious, like low temperatures and freezing water. Have you never heard about the occasional NASA launches that were delayed over the years because of frost warnings in Florida? That would be a bigger problem in North Dakota, that can get close to -40 C in the winter, wouldn't it? Also note that the silly idea you have that cooler minds can work in a cooler environment is just stupid. Have you heard of air conditioning? Do you really think that a mind works better when the outside winter temperature is -30 C in N.D. rather than +25 C in Florida? Phhht! And also, there is no requirement that the engineering and design work be done anywhere near the launch site, right? BTW, did you know that the Challenger shuttle disaster on January 28, 1986 failure of an O-ring seal in the right solid rocket booster? Those O-rings were engineering by Morton Thiokol, which was done in the North (I think it was Illinois, but I am not certain). And one more thing. the o-rings failed because they lost their resiliency due to the low temperature at the time of the flight. So North Dakota would be a stupid place to launch from (unless you are talking about ICBMs). IN any case, do you understand the physics of launch and orbit enough to admit that the costs are lower in terms of fuel weight? === Subject: Re: quiz question for graduate students of math <9K6dneiJDJgfmrbeRVn-3Q@whidbeytel.com> One of the reasons to bring this issue up, of the USA launch sites of the space program in Florida and Texas is that if we build Earth Air-Conditioner of Aluminium sequin placed in orbit around the equator, whether that sequin makes the launch site of Florida and Texas obsolete. The sequin band will be concentrated at the Equator and so rocket launch sites may have to be moved to latitude of 45 degrees north. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: society should use base 16, not base 10 * You can calculate individual digits of PI base 16 without calculating >> the previous digits. Can't do that base 10. > Is this true? By coincidence I was trying to find out about this > recently. I thought I remembered reading somewhere that it had been > achieved for base-10, but I could be mistaken? > In 1995 Bailey, Borwein, and Simon Plouffe (University of Quebec) discovered > some nifty number theory that allows you to calculate individual digits of > pi without having to compute any of the preceding digits for base 2 and 16, > but not for base 10. I have not heard of any method that works in base 10. A few years later Plouffe also published a method for base 10. It has not attracted as much attention as the base-16 (or any other power of 2) method because it's slower than existing methods that compute all preceding digits. It does however have the advantage that it needs *much* less storage for the intermediary results: logarithmic in terms of the digit index, instead of linear (or worse?) for traditional methods. Michel. === Subject: Re: society should use base 16, not base 10 * You can calculate individual digits of PI base 16 without calculating >> the previous digits. Can't do that base 10. Is this true? By coincidence I was trying to find out about this > recently. I thought I remembered reading somewhere that it had been > achieved for base-10, but I could be mistaken? > In 1995 Bailey, Borwein, and Simon Plouffe (University of Quebec) discovered > some nifty number theory that allows you to calculate individual digits of > pi without having to compute any of the preceding digits for base 2 and 16, > but not for base 10. I have not heard of any method that works in base 10. > A few years later Plouffe also published a method for base 10. It has > not > attracted as much attention as the base-16 (or any other power of 2) > method > because it's slower than existing methods that compute all preceding > digits. > It does however have the advantage that it needs *much* less storage > for the > intermediary results: logarithmic in terms of the digit index, instead > of > linear (or worse?) for traditional methods. === Subject: Re: society should use base 16, not base 10 > Unless he's worried about decimal causing cancer. If we thought base 2, we'd hope to live to age 128 rather than hoping to live to age 100. And age 64 would be seen as much more significant. DNA is base 4. Protein encoding, base 64. Base 2 seems more relevant to cancer than base 10. But that's the point, base 2 is more relevant to just about everything than base 10. How would English best name the numbers if we decided to count base 16? 10..15 ought to be single-syllable names that are easily distinguishable from one..nine. I suppose A B C D E F works. === Subject: Re: society should use base 16, not base 10 > Unless he's worried about decimal causing cancer. > If we thought base 2, we'd hope to live to age 128 rather than hoping > to live to age 100. Don't you mean 10000000? > And age 64 would be seen as much more significant. Not really. > DNA is base 4. Protein encoding, base 64. Base 2 seems more relevant > to cancer than base 10. But that's the point, base 2 is more relevant > to just about everything than base 10. But carbon rings are base 6, so go figure. > How would English best name the numbers if we decided to count base 16? > 10..15 ought to be single-syllable names that are easily > distinguishable from one..nine. I suppose A B C D E F works. Now if we can just get those damn C programmers to get with the program. === Subject: Re: society should use base 16, not base 10 > There was a thread on how society should switch from base 10 to base 16 > back in 2001, but somehow nobody brought up the good arguments. Base > 16 is really base 2 grouped conveniently. Google won't let me reply to > it directly, so here's a new thread. Seems to me there you should do one of two things before using base 16: 1. Grow 6 extra fingers; or 2. Cut off a combination of four toes/fingers. --Ron Bruck === Subject: Re: society should use base 16, not base 10 <150920052129031375%bruck@math.usc.edu Seems to me there you should do one of two things before using base 16: > 1. Grow 6 extra fingers; or > 2. Cut off a combination of four toes/fingers. Not counting the thumbs does the trick. Well, that's base 8, but that's another power of 2, which is close enough. Toes don't seem relevant to counting because most people can't fold over individual toes the way the can individual fingers. I can wiggle my big toe independently, and somewhat make my little toe move left and right, but the rest of the toes move in unison. Counting to 1023 on my fingers (or 31 on one hand) is something I can do, but little children can't. They can't control their fingers that precisely. If they learn to count on their fingers, they can only do it by tallying their fingers. === Subject: Re: society should use base 16, not base 10 <150920052129031375%bruck@math.usc.edu Seems to me there you should do one of two things before using base 16: > 1. Grow 6 extra fingers; or > 2. Cut off a combination of four toes/fingers. > Not counting the thumbs does the trick. Well, that's base 8, No, it's base 9. We count from 0, remember? Your first mistake was thinking that finger counting was base 10. It's not. Finger counting is tallying and tallying is not base anything. Because, unless qualified, base n means Standard Positional Number System of Radix n where standard means the symbols start with 0, positional means each digit a(i) represents the value a*n**i (i being the position relative to the radix point) and radix n means there are n symbols. Finger counting would be a Non-Standard Non-Positional Number System of Radix 1 Radix 1 because there is only one symbol 1. Non-Standard because there is no zero, which means it MUST be Non-Positional since the numbers cannot have zero placeholders. Thus, you cannot do standard arithmetic in such a system because you cannot carry past the end of the operands. In standard arithmetic, you CAN carry because, being positional, each number has an infinite number of insignificant 0's past the most significant digit. Numbers in the Tally System of Radix 1 have NOTHING past their most significant digit (because if they did, it must be a 1 since that is the only symbol in the system). And a number can't be added to NOTHING, it can only be added to another number. Now, you CAN do arithmetic, it just has to be non-standard. For instance, you do addition by concatenation: 111 + 11 = 11111. Tally systems are legitimate counting systems, but don't making the common mistake of thinking they are base 1. > but > that's another power of 2, which is close enough. > Toes don't seem relevant to counting because most people can't fold > over individual toes the way the can individual fingers. I can wiggle > my big toe independently, and somewhat make my little toe move left and > right, but the rest of the toes move in unison. > Counting to 1023 on my fingers (or 31 on one hand) is something I can > do, but little children can't. They can't control their fingers that > precisely. Not necessarily. Hold your palm flat with fingers outstretched over the table about a half inch above the surface. Now you can simply raise and lower the individual fingers, no need to curl them. It only takes a little practice and you'll be counting in binary just like a computer. > If they learn to count on their fingers, they can only do > it by tallying their fingers. And not understanding what they are doing and never learning the truth either. === Subject: Re: society should use base 16, not base 10 > There was a thread on how society should switch from base 10 to base 16 > back in 2001, but somehow nobody brought up the good arguments. Base > 16 is really base 2 grouped conveniently. Google won't let me reply to > it directly, so here's a new thread. > * Learning addition and multiplication in grade school would go much > faster. Start with binary. 0+0=0, 0+1=1, 1+0=1, 1+1=10. 0*0=0, > 0*1=0, 1*0=0, 1*1=1. Then define base-4 in terms of binary, and > base-16 in terms of base-4. Learning arithmetic would be heavier on > learning how to carry than memorizing what 5+8 and 6*9 are. > * Base 2 gives order-of-magnitude descriptions more precision than base > 10. > * You can calculate individual digits of PI base 16 without calculating > the previous digits. Can't do that base 10. So 16 is more natural, > at least from PI's perspective. > * Written base 16 numbers are shorter than base 10, but still readable. > * Music (sound, harmonics) is base 2. > * You can count to 1023 on your fingers base 2 (OK, that's not a good > reason.) > * If we don't switch now, we're likely to push a layer of > binary-to-decimal conversion into software and hardware forever. > Imagine a nanotech artificial intelligence traversing the stars 10,000 > years from now, using decimal, Unicode, and x86 instructions in all its > internal workings. With a qwerty keyboard, of course. And beneath all the layers of the operating system, there will still be a DOS window ... === Subject: Re: society should use base 16, not base 10 > There was a thread on how society should switch from base 10 to base 16 > back in 2001, but somehow nobody brought up the good arguments. Base > 16 is really base 2 grouped conveniently. Google won't let me reply to > it directly, so here's a new thread. * Learning addition and multiplication in grade school would go much > faster. Start with binary. 0+0=0, 0+1=1, 1+0=1, 1+1=10. 0*0=0, > 0*1=0, 1*0=0, 1*1=1. Then define base-4 in terms of binary, and > base-16 in terms of base-4. Learning arithmetic would be heavier on > learning how to carry than memorizing what 5+8 and 6*9 are. * Base 2 gives order-of-magnitude descriptions more precision than base > 10. * You can calculate individual digits of PI base 16 without calculating > the previous digits. Can't do that base 10. So 16 is more natural, > at least from PI's perspective. * Written base 16 numbers are shorter than base 10, but still readable. * Music (sound, harmonics) is base 2. * You can count to 1023 on your fingers base 2 (OK, that's not a good > reason.) * If we don't switch now, we're likely to push a layer of > binary-to-decimal conversion into software and hardware forever. > Imagine a nanotech artificial intelligence traversing the stars 10,000 > years from now, using decimal, Unicode, and x86 instructions in all its > internal workings. With a qwerty keyboard, of course. And beneath all the layers of the > operating system, there will still be a DOS window ... Not on Macs or UNIX bases systems. === Subject: Re: society should use base 16, not base 10 back in 2001, but somehow nobody brought up the good arguments. Base > 16 is really base 2 grouped conveniently. Google won't let me reply to > it directly, so here's a new thread. * Learning addition and multiplication in grade school would go much > faster. Start with binary. 0+0=0, 0+1=1, 1+0=1, 1+1=10. 0*0=0, > 0*1=0, 1*0=0, 1*1=1. Then define base-4 in terms of binary, and > base-16 in terms of base-4. Learning arithmetic would be heavier on > learning how to carry than memorizing what 5+8 and 6*9 are. * Base 2 gives order-of-magnitude descriptions more precision than base > 10. * You can calculate individual digits of PI base 16 without calculating > the previous digits. Can't do that base 10. So 16 is more natural, > at least from PI's perspective. * Written base 16 numbers are shorter than base 10, but still readable. * Music (sound, harmonics) is base 2. * You can count to 1023 on your fingers base 2 (OK, that's not a good > reason.) * If we don't switch now, we're likely to push a layer of > binary-to-decimal conversion into software and hardware forever. > Imagine a nanotech artificial intelligence traversing the stars 10,000 > years from now, using decimal, Unicode, and x86 instructions in all its > internal workings. With a qwerty keyboard, of course. And beneath all the layers of the > operating system, there will still be a DOS window ... > Not on Macs or UNIX bases systems. Or Windows XP. === Subject: Re: society should use base 16, not base 10 * You can calculate individual digits of PI base 16 without calculating >> the previous digits. Can't do that base 10. > Is this true? By coincidence I was trying to find out about this > recently. I thought I remembered reading somewhere that it had been > achieved for base-10, but I could be mistaken? > In 1995 Bailey, Borwein, and Simon Plouffe (University of Quebec) discovered > some nifty number theory that allows you to calculate individual digits of > pi without having to compute any of the preceding digits for base 2 and 16, > but not for base 10. I have not heard of any method that works in base 10. I'm probably getting confused then... but if it's possible for a base that is a power of 2 then is it not possible for ANY base that is a power of 2 (i.e. base 4, base 8, base 16, base 32 etc., not just base 2 and base 16)? === Subject: Re: society should use base 16, not base 10 > * You can calculate individual digits of PI base 16 without calculating > the previous digits. Can't do that base 10. >> Is this true? By coincidence I was trying to find out about this >> recently. I thought I remembered reading somewhere that it had been >> achieved for base-10, but I could be mistaken? >> In 1995 Bailey, Borwein, and Simon Plouffe (University of Quebec) discovered >> some nifty number theory that allows you to calculate individual digits of >> pi without having to compute any of the preceding digits for base 2 and 16, >> but not for base 10. I have not heard of any method that works in base 10. >I'm probably getting confused then... but if it's possible for a base >that is a power of 2 then is it not possible for ANY base that is a >power of 2 (i.e. base 4, base 8, base 16, base 32 etc., not just base 2 >and base 16)? Yes, since you can convert directly from base 2 to any base which is a power and 2 by grouping digits. and to convert back simply ungroup. quasi === Subject: Re: Baseball 8. john_rams...@sagitta-ps.com Sep 15, 1:14 pm show options Local: Thurs, Sep 15 2005 1:14 pm === Subject: Re: Baseball Reply | Reply to Author | Forward | Print | Individual Message | Show original | Report Abuse > IF there is a single answer, ie independent to any other > factors, then the scaling MUST logically be linear (otherwise > there _couldn't_ be a single answer). I'm afraid you're thinking of polynomial roots (or something?) What if the OP had said 300' was the maximum _height_ reached by the ball in the first hit? That question would also have a single answer ... Reply Basically I, the OP, mean this: A human batter (or an android, perhaps on the line of Data in STTNG, or even more advanced) on a level field on Earth hit a baseball under normal outdoor conditions that resulted in a line drive that touched down in fair territory 300' away from the plate after the bat hit it at X mph. How high it went, what the barometric pressure is, the angle of the bat, the speed of the ball, the wind resistence, etc. etc. is unknown. Can it be calculated how far the ball would have gone had the bat speed been 2 X mph. I presume it might be less than 600' , but how much less? If the bat had gone at 10 X mph I presume it would be much less than 3000'. I am thinking about writing a sci-fi or fantasy story where a human (or an android who can pass) somehow becomes able to swing a bat many times faster than anyone has done so before and I am doing some research. If even knowing the bat speed was X and the flight of the ball was 300' is not near enough in predicting the results explain why to me please in simple languge:) === Subject: Re: Baseball >Basically I, the OP, mean this: A human batter (or an android, perhaps >on the line of Data in STTNG, or even more advanced) on a level field >on Earth hit a baseball under normal outdoor conditions that resulted >in a line drive that touched down in fair territory 300' away from the >plate after the bat hit it at X mph. How high it went, what the >barometric pressure is, the angle of the bat, the speed of the ball, >the wind resistence, etc. etc. is unknown. Can it be calculated how far >the ball would have gone had the bat speed been 2 X mph. No it can't. It's a rather complicated function of the bat speed, speed of the ball, wind resistance etc., not simply proportional to bat speed, and all you have is one data point. To see it's not just proportional to bat speed, note that if the bat speed was 0 (a poorly executed bunt) the ball would still go some positive distance. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Baseball Suppose a batter hits a line drive 300' after swinging > the bat so that it is going x mph at the point of impact > with the ball, how far would the ball have gone, ALL OTHER > FACTORS BEING EQUAL, if the bat had been going 2x mph? > I reckon the range also doubles, to 600', assuming the ball > leaves the bat at these speeds on both hits, and neglecting > air resistance. > If the guy hits it at the same angle to the horizontal > both times, then the vertical component of the velocity, > say v m/sec in the first hit, also doubles. > Even easier: Equating to zero the height, v.t - g.t^2/2, > at time t gives a flight time of 2v/g (neglecting the height > above the ground of the initial strike point), and the range > is this multiplied by the horizontal component, v_h, of > initial velocity, which is constant. .. constant for a given starting ball velocity but, like the vertical component, v_v, of the initial velocity, doubles when the ball velocity is doubled. So as Steve Mayer's formula shows, the range is 2.v_v.v_h/g and the range increases by a factor of 4. (Same thing applies to the long-winded parabola method: The constant of proportionality of range squared versus v_v squared increases by a factor of 4, over and above the factor of 4 already introduced by doubling v_v.) DOH! Now where did I put that dunce's cap? === Subject: Re: Physics problem but a quetion related to math of it. > In one of the example problems in my book it shows a graph for the > fraction of a complete period that a simple harmonic oscillator spends > within a small interval delta*x at a postion x. Now, I am wondering how > to get this equation because as I work through it I run into a brick > wall with the math. Below is my futile attempt at trying to find it, > We know energy is conserved so that means, > (Total Energy)=(Potential)+(Kinetic Energy) > 1/2*k*A^2=1/2*m*v^2+1/2*k*x^2 (k = spring constant, A = max amp.) > moving things around produces, > sqrt[k/m]*sqrt[(A^2-x^2)]=v=dx/dt > moving things around again gives us, > dt=dx/(sqrt[k/m]*sqrt[(A^2-x^2)]) > Now, if we integrate the above from x to x+delta*x we get, > tdelta=sqrt[m/k]*(ArcSin[(x+delta*x)/A]-ArcSin[x/A]) > since period=2*Pi/w, where w is the angular frequency which is equal to > w=sqrt[k/m] we have, > tdelta/period=1/(2*Pi)*[ArcSin[(x+delta*x)/A]-ArcSin[x/A]] As I said in my post of a few moments ago, this needs to be multiplied by the number of times (i.e. 2, when delta_x is small) that the weight passes through this region per period. > However, this solution makes no sense to me what so ever. What did I do > wrong? Because I have a feeling I am handling the dealta*x wrong. > If the problem involves a pendulum, then x can't be too big; in that > case, you can approximate sin x by x and hence Arcsin(x) by x as well. > The stuff in the brackets turns into x*delta/A. HTH. That's true, but nobody said anything about a pendulum, so I don't think we're justified in making that further simplification. (OTOH if the graph in the OP's book is a straight line, which I doubt, we could conclude that such an approximation was made.) Note, a propos my other postings, the first-order Maclaurin expansion of (A^2-x^2)^(-1/2) is indeed x/A, in other words my answer agrees with yours for small x. === Subject: Re: Physics problem but a quetion related to math of it. being that clear because my quetstion was not clear at all and I was suprised you were able to understand so well. === Subject: Re: Physics problem but a quetion related to math of it. > Could you expand on this statement you made, If it's small enough, you > can skip the integration and just multiply your integrand by delta x. > (This is the same as expanding your general solution to > first order.)? I am a little lost as to why you can do this and how it > works. > Sorry, I could have made that clearer. [snip my clarification] Something I forgot here is that in calculating the fraction of time that the weight spends between x and delta_x, you need to include the fact that it passes point x *twice* in each oscillation (if |x| =/= A, that is). This is an easy correction, of course. If I'd made any real howlers in my postings, no doubt we'd have gotten some corrections by now, so I take the lack of same as a somewhat encouraging sign. But as I said earlier, I still could be wrong. === Subject: Re: How many games of chess are there? >There are those who might argue that the whole thread was just >intellectual masturbation in the first place... > Then stop reading the posts. I am sure that your newsreader > has a killfile as well as a unsubscribe feature. > If, by some chance, you are tied to a chair with your eyelids > taped open and a monitor with a Usenet feed on it in front of > you, then I will address this message to your captors: Firstly, > keep up the good work. Secondly, please take away his keyboard. Maybe he likes intellectual masturbation! Anyway, I think he was joking. You might need to have your humor circuits checked. --Harold Buck I used to rock and roll all night, and party every day. Then it was every other day. . . . -Homer J. Simpson === Subject: Re: How many games of chess are there? >Anyway, I think he was joking. You might need to have your humor >circuits checked. I don't like jokes that are subtle putdowns of others. === Subject: Re: How many games of chess are there? > There are some interesting consequences from the above facts. > It turns out that a powerful enough computer can still solve > chess! Those infinitely long games consist of a finite number > of positions[1], and the computer can stop searching any sequence > of moves when the sequence reaches a position[1] the second time. Well, except that a powerful enough computer is probably physically impossible. In any case, as someone here likes to point out frequently, you don't need to cover every possible game or even every possible position to solve chess, if you can show, for example, that there is a winning sequence for one side or the other. For example, white plays 1. e4. Now you look at all possible black responses, but for each of these you need only show that one of white's counters leads to a win, and so on. Still huge numbers, but less huge. :-) --Harold Buck I used to rock and roll all night, and party every day. Then it was every other day. . . . -Homer J. Simpson === Subject: Re: How many games of chess are there? >Well, except that a powerful enough computer is >probably physically impossible. I wouldn't be so bold as to estimate the probability of a practical quantum computer *ever* being developed. If it takes a thousand years to invent a QC with a couple hundred qbits (more than enough to solve chess), then it will take a thousand years and a day to solve chess, starting today. If it takes a hundred years then it will take a hundred years and a day. A Quantum Computer (if one is ever invented) with enough qbits and a workable algorythm that can solve chess should be able to search for a solution among 2^N possible solutions in N time. A Quantum Computer (if one is ever invented) with enough qbits to solve chess will be able to search for a solution among 2^N possible solutions in N time. A quantum computer that can evaluate 1 position in five milliseconds (my $29.99 LCD handheld can evaluate a positions in one millisecond) would be able to: Evaluate 1 position in 5 milliseconds Evaluate 2 positions in 10 milliseconds Evaluate 4 positions in 30 milliseconds Evaluate 8 positions in 40 milliseconds Evaluate 16 positions in 50 milliseconds Evaluate 32 positions in 60 milliseconds Evaluate 64 positions in 70 milliseconds Evaluate 128 positions in 80 milliseconds Evaluate 256 positions in 90 milliseconds Evaluate 512 positions in 100 milliseconds (0.1 second) ... Evaluate a million (10^6) positions in 0.2 seconds Evaluate a billion (10^9) positions in 0.3 seconds Evaluate a trillion (10^12) positions in 0.4 seconds Evaluate (10^15) positions in 0.8 seconds Evaluate (10^18) positions in 1.6 seconds ... Evaluate (10^30) positions in 25 seconds ... Evaluate (10^36) positions in 100 seconds ... Evaluate (10^72) positions in 200 seconds ... Evaluate (10^108) positions in 5 minutes ...and so on. You can start with quantum computer that can only evaluate 1 position in one second and still solve the game of chess in less than a day. It may turn out that creating such a QC is not possible, but I have never seen proof of that. -- Guy Macon === Subject: Re: How many games of chess are there? <038di1dsrr4tq6eg9614m6hqpodcpbub0t@4ax.com> <4326ad62$0$308$7a628cd7@news.club-internet.fr> <85k6hjvpy8.fsf@lola.goethe.zz> I don't think I've ever seen the Continuum Hypothesis invoked in a >> chess problem before. > (Cymbals Crashing) > NOBODY has ever seen the Continuum Hpothesis invoked in a chess problem > before! > Among our weapons are surprise, fanatical devotion to Gauss... *grrooooooan* Collapses into comfy chair - OH NO!!!! Tim -- When playing rugby, its not the winning that counts, but the taking apart ICQ: 5178568 === Subject: Re: copyright for sudokus ? [fup to rec.puzzles] > (SPOILER for today's NYT crossword follows, by the way.) [...] > Right, but more importantly, if you look at any modern newspaper crossword > (and here I'm talking about American-style), you'll notice > that each one has what crossword designers call a theme, which often > involves wordplay or puns. For example, the NYT crossword today has > the theme Up, with theme entries ALL DRESSED, WHOOPING IT, JOHNNY JUMP, and > WHICH WAY IS. For the record, I didn't even see the whole theme yesterday! The answer today draws attention to the fact that above each of the theme answers in the grid are the letters U and P, e.g.: SIX MONTOYA OPP<--P COTTON BANTU<--U ROTO WHOOPINGIT AZARIA UNSENT JOHNNYJUMP ORGS -Arthur, likes crosswords === Subject: Pi and the integers! What are the rankings of these rationales that are expressed by two integers [n,d] where pi ~ n/d? Ranking is based on total number of digits used between the numerator and denominator and the closest value of pi for that number of digits. In order they are --- (|) denotes separator 22/7 | 333/106 | 355/113 | 103993/33102 104348/33215 | 208341/66317 | 312689/99532 833719/265381 | 1146408/364913 4272943/1360120 | 5419351/1725033 80143857/25510582 | 165707065/52746197 245850922/78256779 .. Each of these in order get closer and closer to the value of pi. I believe these first 14 are the best possible for the total number of digits used and the closest value of pi. Can anyone find a better approximation of pi by matching the total number of digits of any of the above (n,d]? Also for each succeeding n/d the value <>pi, starting with the first n/d is >pi,then the next is pi, pi, pi... This alternating of <>pi continues on for each new n/d. These were found on my 'new' way of finding numerators and denominators on any amount of sequential terms in a cf. So yes, these numerators and denominators were produced from the cf of pi. Dan === Subject: Re: Is math a real science? For some similarities between the empirical sciences and mathematics, see Lakatos Proofs and Refutation CUP.> Mark Demers quoted carelessly: >> proof and refutation And commented uncomprehendingly: > theatre has theatre critics and art has art critics > too. that doesn't > make theatre and art science. Come on, Mark. If you were acquainted with Lakatos, and I would suggest also, Tarski's Correspondence Theory of Truth, you might find some enlightenment on the inner workings of language that make it useful for other than bloviating. And far more interesting as well. Tom === Subject: Re: Is math a real science? ... >On further reflection, I also think that part of my problem with >saying the musician is accumulating knowledge in this case is that >the definition that was at issue spoke of an accumulation of knowledge >not in the sense of a single individual acquiring inherently >untransmittable knowledge, but rather as an activity that increases >the general pool of knowledge accessible (at least in principle) to >everyone. The musician may be learning more, but he is generally >incapable of transmitting this knowledge ot the population at large in >a useful way that would make this knowledge something that population >can use. As such, I would say that even if we call it accumulation of >knowledge, it would be qualitatively different from the activity >being refered to in the definition of science at issue. The distinction being discussed in this thread is, apparently, often called that between knowledge by description and knowledge by acquaintance (terms that seem to have been introduced by Bertrand Russell). It certainly seems clear to me--though no doubt any half-decent philosopher could unclarify things in an instant--that, of the two, knowledge by description is much more easily transmitted from one person to another by the channel of transmission that is the favorite of nearly all of us here (especially insofar as we are, after all, here in a Usenet newsgroup), namely, _language_: for language is, among its other functions, the favorite medium for _description_ (particularly for people who spend lots of time writing, and who are either paid or pay for others to spend lots of time talking). Yet it also seems clear to me that there are many instances where knowledge by acquaintance is also transmitted, and indeed in many of those the use of (verbal, perhaps other) descriptions is of little use. All sorts of physical skills appear to be of this sort, from lovemaking to musicmaking, with juggling and riding the unicycle somewhere in between (depending on your personal style of each, no doubt). In the remote era when I hung out on the edges of the old AI Lab, people in general (well, I mean, people in the general population at MIT, which isn't *quite* representative of the general population _tout simple_) seemed to think that it was odd, dubious, or at least worthy of somewhat skeptical remarks that Seymour Papert and his group (for instance) spent a lot of time demonstrating--or trying to demonstrate--that various kinds of knowledge-by-acquaintance (including, from the list above, juggling and unicycling, as well as several other circus arts) *could* in fact be supplemented, if not subsumed, by knowledge-by-description. Margaret Minsky got a bit of early celebrity for writing an undergraduate paper on How to talk about jigsaw puzzles as if they were important (I may have the title wrong, but the concept is right), the point (or one of the points) being that lots of master jigsaw assemblers don't talk about what they're doing, yet apparently can transmit their mastery to apprentice jigsaw assemblers. The traditional master/journeyman/apprentice system in the trades (less so in the professions) is a codification of the apparent fact that knowledge transmission can happen largely without verbal instruction. Lee Rudolph === Subject: Re: Is math a real science? > Physics and biology are sciences because they put > theories up to > falsifiable tests. > But most of the esoteric math theorems in abstract > set, algebra, > geometry, and field > theory have not been tested. Moreover, they cannot > be tested. Thus, > they cannot be > scientifically verified or falsified. > Is math, thusly, a pseudo science? > Lunk Is all science empirical? Tom === Subject: Re: Is math a real science? >>Physics and biology are sciences because they put >>theories up to >>falsifiable tests. >>But most of the esoteric math theorems in abstract >>set, algebra, >>geometry, and field >>theory have not been tested. Moreover, they cannot >>be tested. Thus, >>they cannot be >>scientifically verified or falsified. >>Is math, thusly, a pseudo science? >>Lunk > Is all science empirical? Maybe we should divide the sciences into two classes: 1. Empirical where experimental results decides soundness. 2. Non-empirical where deductive correctness decides soundess. Mathematics would be in category 2 and not in category 1. Bob Kolker === Subject: Re: Is math a real science? >>Physics and biology are sciences because they put >>theories up to >>falsifiable tests. >>But most of the esoteric math theorems in abstract >>set, algebra, >>geometry, and field >>theory have not been tested. Moreover, they cannot >>be tested. Thus, >>they cannot be >>scientifically verified or falsified. >>Is math, thusly, a pseudo science? >>Lunk Is all science empirical? > Maybe we should divide the sciences into two classes: > 1. Empirical where experimental results decides > soundness. > 2. Non-empirical where deductive correctness decides > soundess. > Mathematics would be in category 2 and not in > category 1. > Bob Kolker I concur. Mathematics holds in common with the empirical sciences, the properties of theory and result. Sometimes the result is empirical and often it isn't. Also, the deductive process demands that theory is primary (although results, e.g. the empirical observations on which quantum theory was founded, may temporally precede the theoretical explanation). Tom === Subject: Re: infinity Daryl McCullough said: > Tony Orlow says: >Daryl McCullough said: >> That's what I said. For any natural number n, size(A_n) = n, >> and n = the largest element in A_n. That doesn't say anything >> about a set with *no* largest element. >It says that there is no finite n in N for which A_n is infinite. > That's right. >If no initial segment is infinite, how can the set be infinite? > Because the set of all finite naturals is larger than any proper > subset. So the size of the set of finite naturals is larger than > any finite natural. Infinitely larger? Is the set more than 1 larger than every proper subset? Can that ever be the case, that you cannot remove 1 element and have a proper subset? > The set of all finite natural numbers does not have that form. >>It is the complete initial segment of itself, the non-proper subset. >> But it has no largest element, so its size is not equal to its >> largest element. >Every element is a finite natural, > True. >and for no finite natural does there exist >an infinite set of predecessors. > True. >Therefore the set is not infinite. > The therefore doesn't follow. The set is *larger* than any of > its proper subsets. So the fact that each proper subset is finite > does *not* imply that the entire set is finite. Is there a proper subset with 1 element missing? > The size of the set must be larger than the size of any of its > proper subsets. The only way a size can be larger than any finite > size is to be infinite. The proper set with 1 element missing is smaller than the set, by 1 element. If that proper subset is finite, can adding 1 element make it infinite? > -- > Daryl McCullough > Ithaca, NY -- Smiles, Tony === Subject: Re: infinity > Daryl McCullough said: >> Tony Orlow says: >>Daryl McCullough said: > That's what I said. For any natural number n, size(A_n) = n, > and n = the largest element in A_n. That doesn't say anything > about a set with *no* largest element. >>It says that there is no finite n in N for which A_n is infinite. >> That's right. >>If no initial segment is infinite, how can the set be infinite? >> Because the set of all finite naturals is larger than any proper >> subset. So the size of the set of finite naturals is larger than >> any finite natural. > Infinitely larger? Is the set more than 1 larger than every proper subset? Can > that ever be the case, that you cannot remove 1 element and have a proper > subset? It is infinitely larger than every finite proper subset. You cannot remove a single element from the set of all finite naturals and have a finite set. Remember, your proof was only about finite sets of finite naturals. You draw conclusions about infinite sets of finite naturals from a proof about finite sets of naturals. >> The therefore doesn't follow. The set is *larger* than any of >> its proper subsets. So the fact that each proper subset is finite >> does *not* imply that the entire set is finite. > Is there a proper subset with 1 element missing? There are an infinite number of them, and they are all infinite. Note, none of them have the form you have been talking about. If I take the set of all finite natural numbers, and remove one element, I will never have a set of the form { 1, 2, 3, ....... n } for some finite n. Stephen === Subject: Re: infinity Because the set of all finite naturals is larger than any proper > subset. So the size of the set of finite naturals is larger than > any finite natural. > Infinitely larger? Yes. Given any finite number, there are infinitely many finite numbers which are greater. > Is the set more than 1 larger than every proper subset? Does the set more than 1 mean the set of naturals greater than 1? I don't understand this question. > Can > that ever be the case, that you cannot remove 1 element and have a proper > subset? If you remove one element from N, you have an infinite subset which is a proper subset of N. > The proper set with 1 element missing is smaller than the set, by 1 element. > If > that proper subset is finite, can adding 1 element make it infinite? That proper set is not finite. - Randy === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> Daryl McCullough said: >> Because the set of all finite naturals is larger than any proper >> subset. So the size of the set of finite naturals is larger than >> any finite natural. >> Infinitely larger? > Yes. Given any finite number, there are infinitely many > finite numbers which are greater. >> Is the set more than 1 larger than every proper subset? > Does the set more than 1 mean the set of naturals greater > than 1? I don't understand this question. >> Can that ever be the case, that you cannot remove 1 element and >> have a proper subset? > If you remove one element from N, you have an infinite subset which > is a proper subset of N. >> The proper set with 1 element missing is smaller than the set, by 1 >> element. If that proper subset is finite, can adding 1 element >> make it infinite? > That proper set is not finite. Well, you have overlooked that the sentence from Daryl that Tony has focused on is >> Because the set of all finite naturals is larger than any proper >> subset. And indeed, this sentence as a statement of set size is simply wrong. And not only that, Tony's counterargument actually seems to have focused on why it is wrong, even though it is somewhat cloaked by his generally fuzzy use of language. So I am quite of the opinion that Tony has scored a valid point against Daryl. It's not like Tony has an abundance of valid points, so we should not become envious because of this one. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity Randy Poe said: > Randy Poe said: > I've already posted the relevant textbook quotes. > Yes, and what you claimed he was saying he never said, > Of course he did. I posted the relevant quotes. > and in fact, he said sum > (x->oo: f(x)). > he described it as a collection of marks or symbols without inherent > meaning. > He then went on to say that the meaning of this set of marks > scratched on paper, is the limit of a sequence of finite > partial sums, just as I did. > On the previous page he explains, in quotes I also provide, > that the limit of a sequence is a description of how the > finite values behave at finite indexes in the sequence. > I know what an infinite series is, and your claim that a sum of > an inifinite number of 1's is a finite number is ridiculous. > Where did I say that? > I said that in concluding that the meaning of the notation > sum(k:1,oo)a_k is not a finite number, what is meant is > that the finite values taken by sum(k=1,n)a_k have the > property that they will eventually exceed any finite > number you pick. > Therefore we write another series of marks, the limit > of the sequence is infinite. > The point is that we never claim we have added up an > infinite number of terms or gone to infinity or > examined the infinitieth term. We look at how the > finite partial sums behave, and if we find they grow > without bound then we say the go to infinity even though > every finite partial sum has a finite value. > It is very clear that you have no clue what I just said > in the paragraph above and instead read it as > blah blah blah finite blah blah blah finite blah > blah blahblahblah finite. That's the only way I can > figure how you can go from the limit of the sequence > of finite partial sums is not a finite number to > the sum of the infinite series is infinite. > - Randy I go from the one to the other thus: the limit of the sequence of finite partial sums=sum of the infinite series and is not a finite number=is infinite I understand the method with partial sums, but what it figures is the sum of the infinite series of terms, at least if that converges. -- Smiles, Tony === Subject: Re: infinity David R Tribble said: > Let's look at the generation of the naturals another way. We start with an > empty set, then perform the following steps: > 1. Increment all values in the set by one. > 2. Add the element 1 to the set. > 3. Go to step 1. > Now, the largest value in the set at any given time is the first one added in > step 2 of iteration 1. It is always the same element, and we know it exists, > since we added it first. At any given point, at iteration x, it is equal to 1 > incremented x-1 times. Now, will this first element be finite, in an infinite > number of iterations, after being incremented an infinite number of times, to > generate an infinite number of successors? Remember, an ordered set doesn't > HAVE to have ascending order of value, just some linear order. > This algorithm doesn't generate the set of all naturals. It generates > an infinite number of sets, each one larger than the previous one, > and each one containing a finite number of finite naturals. > We can name them, if you like: > S_0 = {} > S_1 = {1} > S_2 = {1, 2} > ... > S_k = {x : x <= k}, for all finite natural k > 0 > Each set S_k contains k members, and its largest member is k. > We get an infinite number of sets, but none of them is an infinite set. > Isn't this fun? Funny maybe. For each set you generate, you have one more lement. Set 9 has 9 elements. Set zillion has a zillion elements. If you generate oo sets, set oo has oo elements. Now THAT'S fun! -- Smiles, Tony === Subject: Re: infinity > David R Tribble said: >> Let's look at the generation of the naturals another way. We start with an >> empty set, then perform the following steps: >> 1. Increment all values in the set by one. >> 2. Add the element 1 to the set. >> 3. Go to step 1. >> Now, the largest value in the set at any given time is the first one added in >> step 2 of iteration 1. It is always the same element, and we know it exists, >> since we added it first. At any given point, at iteration x, it is equal to 1 >> incremented x-1 times. Now, will this first element be finite, in an infinite >> number of iterations, after being incremented an infinite number of times, to >> generate an infinite number of successors? Remember, an ordered set doesn't >> HAVE to have ascending order of value, just some linear order. >> This algorithm doesn't generate the set of all naturals. It generates >> an infinite number of sets, each one larger than the previous one, >> and each one containing a finite number of finite naturals. >> We can name them, if you like: >> S_0 = {} >> S_1 = {1} >> S_2 = {1, 2} >> ... >> S_k = {x : x <= k}, for all finite natural k > 0 >> Each set S_k contains k members, and its largest member is k. >> We get an infinite number of sets, but none of them is an infinite set. >> Isn't this fun? > Funny maybe. For each set you generate, you have one more lement. Set 9 has 9 > elements. Set zillion has a zillion elements. If you generate oo sets, set oo > has oo elements. Now THAT'S fun! There is no set oo. Presumably that would be the last set, and there is no last set in Z. Now I can already hear your objection: if there is no oo element, then the set is not infinite Well, that may be true according to your definitions, but it is not true according to any standard definitions. In any case, it is not true of your definitions either. Consider the set of TOnats { 1, 2, 3, ....... oo-2, oo-1 } This set does not contain an oo element. There is no ooth element in the set. It only has oo-1 elements. However I am quite sure you think it is infinite. Stephen === Subject: Re: infinity David R Tribble said: > David R Tribble said: >> To be fair, we've all listened with an open mind. It's just that your >> arguments are flawed and therefore not persuasive. > Look, if after all this explaining you are unable to grasp ANY of my points, > then indeed you are NOT reading with anything like an open mind and your > claim to the contrary is simply not true. > Having an open mind means that we're open to new ideas and are willing > to listen to them. We've obviously done that; the length of this > thread attests to that. But having an open mind does not mean we have > to accept those new ideas. Indeed, the onus is on you to prove to us > that these new ideas are true. Indeed the length of the thread attest to something, though it seems some, like Virgil, spend their time trying to shoot my ideas down, rather than ever considering them. Others have made attempts to understand, and maybe even gotten a glimmer of what I am doing, but it certainly feels like anything outside of accepted mainstream thought is automatically considered crankhood. I don't expect you to accept what I say because I say so. I am not an authority > But your proofs, so far, haven't convinced us, because they are not > well-formed and not consistent. Your intuition approach to > infinite sets does not seem to hold up under logical scrutiny. You have agreed that no finite natural has an infinite number of predecessors, becaue each natural is the size of the set of naturals from 1 through that natural. We are almost there, slow going as it has been. > How can a set where every element has a finite number of predecessors be > infinite? > Because every element has an infinite number of successors. > Zero is less than every element in the set (except itself, of course), > but there is no element that can be named that is greater than every > other element in the set. You've said so yourself, several times. But, if there are an infinite number of successor operations which take us from element x to element y, then there are an infinite number of predecessor operations which take us from element y to element x, and y therefore has an infinite number of predecessors, which we have agreed is impossible in the finite naturals. -- Smiles, Tony === Subject: Re: infinity > But, if there are an infinite number of successor operations which take us from > element x to element y, then there are an infinite number of predecessor > operations which take us from element y to element x, and y therefore has an > infinite number of predecessors, which we have agreed is impossible in the > finite naturals. But there is no element y that requires an infinite number of successor operations to reach it from x. For any two finite x and y, x-y is finite. But there are an infinite number of different x and y, and there is no finite n such that n > x-y for all x and y. You are again assuming your conclusion in your proof. You are trying to prove that a infinite set must contain infinite elements, and your argument is using the fact that an infinite set must contain infinite elements. Stephen === Subject: Re: infinity element x to element y, There aren't. There are an infinite number of successor operations, and each one takes you to a different, but still finite number. Imagine I step through the finite numbers, one per second. You agree that this stepping will never end, that in each second I will still be at a finite number and I will never run out of steps. I can continue stepping through finite numbers forever, one per second. How many time steps do you think there are in forever? - Randy === Subject: Re: infinity David R Tribble said: > David R Tribble said: >> What is your definition of an infinite set? I know, I know, it's a >> set with an infinite number of elements. Or unbounded number of >> elements, whatever. But how do you know the number of elements is >> really infinite? > By the properties of the elements and the construction of the set based on > those properties. Unboundedness doesn't mean infiniteness. If the set has an > infinite range of value and a finite or infinite number of elements per unit > of value, or has a finite range of value and an infinite number of elements > per unit of value, then it is has an infinite number of elements. > So how do I know what an infinite range is? How big does a range > have to be to be infinite? Can I just compare it to, say 'N', or > perhaps log(N)? The range is the maximum difference bwteen any two values. If all values are finite, then all difference are finite, and so is the range. AN infinite range of values means there are two values which have an infinite difference. If one is finite, the other must be infinite, since any two finites only have a finite difference. They may both be infinite, and still have an infinite difference between them, but they cannot both be finite. > Or how do I count a set to know that it has an infinite number of > elements? Obviously, I can't use the counting numbers to count them > (ironically) because you've said that there are only a finite number > of them, so how do I enumerate the members of a set to find out if > the set is finite or infinite? If the set is defined recursively without bound it is infinite by nature. The only reason your set of natural is finite is because you have imposed the restriction of finiteness on its members, thereby restricting the value range to a finite value, and given the fact that each member occupies a finite portion of this range, one can only have a finite number of members within that finite range. Now, if you want to enumerate the elements of the infinite set, you need to denote each element by some infinite representation: an infinite quantity, string, etc. Enumerating an infinite set using finite representations will not work. -- Smiles, Tony === Subject: Re: infinity David R Tribble said: > Even your son apparently understands that. > Actually, I think Tony mentioned his nine-year old nephew. 11 year old son, but whatever....... > Pay attention, okay? You're never going to appreciate the genius of > Orlowian math if you don't pay attention! That's right!!! :D > It's all quite simple. N is the set of naturals, which includes > infinite naturals, not to be confused with N, the old definition of > the set of naturals which does not include infinite naturals. > Then there's N, which is the size of N (the new set N, not the > old set N). N (the number) is a natural number, an infinite natural, > and is the basic unit of infinity (the unbounded kind of infinity, > not the bounded kind). And N (the number) is a member of N (the set, > that is, the new set), because N (the set) contains all the infinite > naturals. But N (the number) is not a member of N (the old set) > because every finite ordered set contains its size as its greatest > element, and the size of N (the old set) is a finite number less than > N (the number). But size of N (old set) is unidentifiable, but > definitely finite, because N (the set) is finite. N (old set) is a > finite set, of course, because it contains only finite numbers. > But N (the new set) is an infinite set, because it contains all the > naurals, even the infinite ones. And since N (the number) is infinite, > -N, N-1, N+1, N/2, sqrt(N), log(N), etc., are all infinite numbers. > It's all so simple that a nine-year old can follow it, see? > Since so very few of us can ever get it all straight must mean that > we're all stupider or something. If you were to point out a contradiction within what I am saying, without the standard assumptions, that would be a worthwhile objection. However, I don't see any contradictions above. COuld you point them out, perchance? > Here's a TOmatics puzzle I have not been able to figure out. > You claim that oo (or N, or whatever) is infinite, and that > it has an infinite number of predecessors. What are those > predecssors? Well there are the finite numbers, but you > say there are only a finite number of those. There are > according to you oo-1, oo-2, oo-3 and so on, but again > there are only a finite number of those, because there are only > a finite number of finite numbers. There is also oo/2, oo/3, etc > but again there are only a finite number of those. You are assuming that those numbers, 1,2,3 etc are all finite, but remember, I am not restricting my naturals to finite values, so you have one continuous set with an infinite number of successors. The Twilight Zone cannot be crossed in finite numbers of finite setps, but infinite umbers of steps can do it. > You're probably forgetting M-N and N/M, where M is any other infinite > natural greater than N. There's an infinite number of them, too. > But if M = N+k, then M-N = k, so there should only be a finite number > of them, too. Or maybe not. Who knows? You appear almost to be actually trying to think about this. That is a good first step. let me try to clarify what I think your question might be. M might be finitely larger than N, or infinitely. It could be N+k, or N+X where X is infinite. So M-N could be finite or infinite, depending. You mention N/M with M>N, which would be less than 1. If M is N+k with k finite, N/M would be infinitesimally less than 1. If M were 2N, N/M would be 1/2. If M were N^2, N/M would be 1/N, or the unit infinitesimal, or as you prefer, zero. > And maybe Orlowian math has an infinite number of as-yet undiscovered > arithmetic operators to draw from as well. Er, ah, no. -- Smiles, Tony === Subject: Re: infinity Jiri Lebl said: >> (A z)(z < x => (E y)(y < z)). That's not true, Take N union {aleph_0} and order it normally. > Quite right. I realized this after posting, but didn't fix it. > I make mistakes like that all the time. My advisor just yesterday > caught me doing a totally freshman algebra mistake (I messed up the > distributive law, because I was so fixated on the result that I didn't > check my work:) Huh! Given your attitude I rather imagined you were older. Just a young 'un I see. Maybe there's hope for you yet! Of course, eberyone makes mistakes. > Anyway, I suppose everybody (except Don Knuth I've heard) makes > mistakes, and unlike Tony those that have some functioning neurons, > realize when they've messed up and admit it. Now there are two > possibilities for Tony. Either he is a total idiot and has not > realized he's wrong, or he's just immature and won't admit he's wrong. Gee, since I have probably been thinking about this longer than you've been alive, you might want to reserve judgement before calling people idiots, just because they disagree with the mainstream. By the way, hard right conservatives think I'm an idiot too, but I really can't be overly concerned with that. > Of course, the circularity is due to Tony's bleatings. He wants to > say that an infinite set must have elements that are infinite in size, > infinitely far apart, have infinitely many predecessors, etc. Maybe > he even thinks that's a definition. But the use of infinite in each > case is either undefined or refers again to set size. > Every possible definition of infinite size I can think of which would > involve having a linearly ordered subset with no end in effect. But > Tony thinks that is finite. Unbounded. > Once you've put two ends on a set, and > don't think that something which doesn't end is infinite. Then I can't > see any possible definition for infinity. Infinity essentially mean unending, and the Peano axioms define an infinite set. But, that set has infinite elements in it, due to the 1-1 correspondence between position and value for each element in the set. > I think Tony's numbers have a sort of structure like > {0,1,2,3,...} union {...,N-3,N-2,N-1,N} {-oo, -oo+1, -oo+2.....-2, -1, 0, 1, 2.........oo-2, oo-1,oo} One CAN cross the Twilight Zone with INFINITE increments. > In that ordering. That is two copies of normal finite natural numbers > pasted next to each other. Though obviously he thinks that these would > be finite sets, so their union must be finite. Thus the set looks > like: > {0,1,2,3,...} union {Twilight Zone} union {...,N-3,N-2,N-1,N} Obviously??? No. If your second set had only finite subtractions from N, it would be finite, and the bottom end would have infinite values as well. See my other post where I compare the interval [0,1] to the interval [1,oo]. The finites, at the infinite scale, are all clustered at the point next to 1. Any finite distance from that end of the line denotes an infinite value. So there are infinitely more infinite values than finite ones. The twilight zone is the point which is the largest finite number of points from 1, the mark between nothing and something on that scale. > Where the twilight zone makes the set infinite, but it is something you > can't quite get at because if you could find and identify any number in > it, you could prove contradictions. You would define the largest finite and smallest infinite, which would indeed produice contradictions. > So you can't detect it in any set, > so I assume you have to use some supernatural powers to feel the > infiniteness of the set. Uh, yeah, the force, or whatever. (sigh) > Somehow the twilight zone must contain most > of the elements of the set, but no matter what operations you do you > can never land there other wise you would get some concrete numbers in > the twilight zone. The twilight zone doesn't contain any elements. The largest finite natural is like the smallest finite real. It can't be determined. > Of course there are other contradiction from this as for example you > will NEVER get out of the first set by just adding 1 so you never get > to the twilight zone and so most definitely you never emerge on the > other end. The moment you touched the Twilight Zone, you would already be on the other side. You cannot get to the other side in any finite number of finite steps. You need an infinite number of steps, or infinite-size steps, to achieve infinity. > Jiri -- Smiles, Tony === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > Gee, since I have probably been thinking about this longer than > you've been alive, you might want to reserve judgement before > calling people idiots, just because they disagree with the > mainstream. Oh, it is exactly _because_ you have been thinking this long about it and still get it completely messed up, that you are worthy of the name idiot. It is completely baffling how you can continue to spew incoherent nonsense for months and months when people keep walking you through your basic mistakes and showing them how you constantly contradict yourself. You really _are_ an idiot, and _exactly_ because you expose yourself to the material for so long, and still, after years, have not managed the understanding of an average junior high school student. It is not that your ideas are hard to grasp, it is merely impossible to grasp more than one at once, since all of them are contradictory. You are not at all bothered about coherency in your own ideas, and that means that even you yourself treat your ideas like the ideas of an idiot: you don't bother remembering them or making them part of a consistent world view. > Infinity essentially mean unending, and the Peano axioms define an > infinite set. But, that set has infinite elements in it, due to the > 1-1 correspondence between position and value for each element in > the set. Whining does not make it so. There are neither infinite position or values in the set. All of them are finite. But there is no limit to the number of finite values and positions, and so range and set size of the naturals are infinite. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity Randy Poe said: > David R Tribble said: > If no element has an infinite number of elements before it, then there are > not an infinite number of elements. If you maintain there are, then please > explain how a set where no element has an infinite number of elements > before it can possibly be infinite. Obviously, because any given (finite) element of the set has an > infinite > number of elements following it (successors). That's one definition of > an infinite set. > Uh, if any element has an infinite number of finite successors, > All do. Nope. > then one of > those finite successors has an infinite number of predecessors, > Nope. Yep. If x is n steps past y, y is n steps before x. Are you suggesting that the set is one size if counting up, and a different size if counting down? And people think I'M non-standard. > This makes your usual assumption, that you can't have an > infinite number of things which are all at finite distance. Not if each occupies a finite portion of that distance, no. > Even though that contradicts your (correct) view that > if you take one step at a time, you will always be a > finite distance from your starting point and will never > run out of steps to take. In a finite number of iterations. > - Randy -- Smiles, Tony === Subject: Re: infinity Tony Orlow says... >> Uh, if any element has an infinite number of finite successors, >> then one of >> those finite successors has an infinite number of predecessors, >> Nope. >Yep. If x is n steps past y, y is n steps before x. As usual, you are taking a true statement: A: If x+n = y, then y-n = x and you are saying it proves a false statement: B: if x has infinitely many successors, then x has infinitely many predecessors >Are you suggesting that the >set is one size if counting up, and a different size if counting down? No. Everybody agrees with A. But they don't agree that A implies B. Why do you believe that A above implies B? -- Daryl McCullough Ithaca, NY === Subject: Re: infinity > Randy Poe said: >> David R Tribble said: >> If no element has an infinite number of elements before it, then there are >> not an infinite number of elements. If you maintain there are, then please >> explain how a set where no element has an infinite number of elements >> before it can possibly be infinite. >> Obviously, because any given (finite) element of the set has an >> infinite >> number of elements following it (successors). That's one definition of >> an infinite set. >> Uh, if any element has an infinite number of finite successors, >> All do. > Nope. >> then one of >> those finite successors has an infinite number of predecessors, >> Nope. > Yep. If x is n steps past y, y is n steps before x. Are you suggesting that the > set is one size if counting up, and a different size if counting down? And > people think I'M non-standard. There is no x and infinite number of steps past y. >> This makes your usual assumption, that you can't have an >> infinite number of things which are all at finite distance. > Not if each occupies a finite portion of that distance, no. 'that distance'?? What is 'that distance'? Randy did not mention a single distance. Again you are making the mistake of equating for all x and y, there exists a finite n such that x-y < n and there exists an n such that for all x and y, x-y < n >> Even though that contradicts your (correct) view that >> if you take one step at a time, you will always be a >> finite distance from your starting point and will never >> run out of steps to take. > In a finite number of iterations. What does 'never' mean to you? Where do you get your definitions from Tony? Do you and Lester both have some direct connection to the God of Definition? Stephen === Subject: Re: infinity stephen@nomail.com said: > stephen@nomail.com said: >> But there is no largest member, right? Or is there? >> I am never sure if you think there is one or not. > There is no largest finite, since for any finite you can name, there is another > one after it. We all know that. The set has no finite upper bound, and there is > no largest natural number. We all agree on that, I think. We differ in what > kind of significance we attribute to the boundlessness of the naturals. > So if the largest finite does not exist, then nothing, including > the set size, equals the largest finite. Something can be the same as something else and not exist, as long as the other thing doesn't exist either. A Flornkney is a purple flying toadstool. They are the same thing. Neither exists. > If you want to say that ordered sets without a largest element > do not have a size then that is reasonable. Claiming that > the size is something that does not exist is unreasonable. Well, that is essentially what I am saying. If there is a size of such a set of naturals starting from 1, then that is also the largest element, and if there is a largest element, then that is the set size, but if we prove there cannot be any largest element, then there cannot be any set size, since that would be equal to the largest element. The largest element and set size are the same number, even if neither exists. You only get a contradiction by equating them if in some case one exists and the other doesn't. Of course, there will eb some responses like, Yes that is the case. Aleph_0 is the set size but there is no largest finite, so you have a contradiction. But, you cannot object using your conclusion as premise. Remember, you are the one claiming that you can count > through all the finite naturals at a rate of one per second > in a finite number of seconds. Nobody else thinks this > is possible. >> That must be because they think there is some finite natural that take an >> infinite number of seconds to count to. Which one is that? >> That is just your quantifier dyslexia talking. > I beg your pardon? If no element takes an infinite amount of time to count to, > then what are you counting after an infinite amount of time? You know what you > can do with your accusations of QD. > I am not counting after an infinite amount of time. I do > not know what you even mean by that. If I start counting > the finite naturals, I will never stop. There is no last > element. For every finite time, there are numbers I will > not have reached yet, and at no finite time will I have > reached every finite number. And at no infinite time will you still eb counting finite numbers. You will count for all finite iterations, but no infinite iterations. >> There >> is no finite natural that takes an infinite number of >> seconds to count to. But you can always keep counting. >> No matter how many seconds you count, there are still >> more finite naturals. If you stop after a finite amount >> of time, there will still be an infinite number of finite >> naturals you have not yet reached. > Yes, you count for all finite time, but you do NOT count for an infinite amount > of time, or you would be counting infinite naturals, not finite ones. > You need to define precisely what you mean by an infinite amount of time. > I guess you want people to count forever, and then somemore. > Do you think there is time after forever? This is one of the > reasons why bringing time into discussions of infinity seems > to hinder more than help. Think of it as times, or iterations. When you count an infinite number of iterations, you are up to infinite values, one unit of value for each iteration. > Anyway, if I count for an infinite amount of time, meaning that > I never stop counting, then I will always be counting finite natural > numbers, and I will never stop counting finite natural numbers. You will count finite numbers for all finite numbers of iterations. Infinite numbers of iterations produce infinite values. So here you seem pretty sure that you can count > all the finite naturals in F seconds[1], where F is the > largest finite natural. Remember, that is your claim, > not anybody else's. >> Answer the question I just asked. Which finite natural takes an infinite amount >> of time to count to? >> None of them do. But for any finite amount of time there >> are naturals you cannot count to in that amount of time. > If you count for any aprticular finite amount of time, there is another finite > second after that, so you have not counted for ALL of finite time. This is your > typical largest finite conflation and confusion. Get over it. Boundless does > not mean infinite. > Tony, you are the one claiming that you can count all of the > finite naturals in a finite amount of time. You seemed to > have changed your position in mid argument. No, I am not changing my position. You see an unbounded set and think it's infinite, but by the properties of the elements, it cannot be. No finite number is infinite, therefore no number in the set has an infinite number of predecessors, and the set is finite. Can you identify the end? No. Does that make it infinite? No. > So once again, do you think that you can count all of the > finite naturals in a finite amount of time if you count one > natural per second? And by count all of them, I mean > all of them, starting with 1, in one interval of finite time. Can I tell when it will end? No. Will it go on for an infinite number of iterations? No. > Note, I do not mean each of them. We all agree that given > a single finite natural, I can count to that finite natural > in a finite amount of time (ignoring my own mortality and > the eventual heat death of the universe). The question is > can I count > 1, 2, 3, 4, .... > and say the name of each and every finite natural number > in a finite amount of time. If you have a finite number of them, and they each take finite time, then it takes finite time. You believe you have an infinite number of them, but an infinite number of increments makes an infinite sum, so you will have infinite values if you use an infinite number of iterations. You need a finite, but unbounded number of iterations. >> Yes, well, that's a consequence of your focus on the finite naturals and their >> unboundedness, which still doesn't make them an infinite set. >> There are infinite by every standard definition of infinite. > I am obviously using non-standard approaches, because the standard approaches, > in my not so humble opinion, suck. > So far your non-standard approaches so far are not very > consistent. And insisting on using a word differently > than its standard meanings is just asking for confusion. Infinity is the right word. The concept needs fixing. Notice I do NOT use cardinality. That is a registered trademark of CantorCo. Infinity is not. I define it differently, and I am allowed to. >> You refuse to share your definition of infinite as it applies >> to sets. I saw all that stuff about 0> you never explained how that applied to sets. > Yes, I defined finite sets as sets with a finite NUMBER of elements, after > defining finite NUMBER. > But you did not describe how you determine the NUMBER of elements > of a set. What is the NUMBER of elements in the set of finite > naturals? You have been asked this question dozens of times, > and I have still yet to see a real answer from you. Then try opening your eyes. I have explained the relationship between this size and the largest member of this set. If you didn't get is the first dozens of times then I don't know what to do for you. There is no distinct size. Claiming there is is a mistake. >> I asked you >> what x was for the finite naturals and you refused to answer. > What x was? What does that question mean? x is ANY finite natural, in the > definition I suggested. > No, you claim that the number of finite naturals is a finite > number. According to your definition, the number of finite > naturals equals 1/x where x>0. What is x? Why should > I believe x is greater than 0, and that 1/x is therefore > finite according to your definition, if you cannot tell > me what x is? Do you understand how definitions work? These are the conditions for determining if ANY number x is finite: 1. 0> If you cannot tell me what x is, how do I know that it is >> greater than 0? > What are you talking about? The largest finite again? I don't understand the > question. > See above. I maintain the set of finites is finite, but that the size cannot be pinpointed, so certainly the reciprocal of the size also cannot be pinpointed. >> And you still have not explained how the number of finite >> naturals can be a finite natural. > That I have explained countless times it seems. Remember the proof that all > sets of naturals starting from 1 have a largest element equal to the set size? > Yep. > If all elements in the set are finite naturals, how can the size be anything > BUT a finite natural? > That is your argument? If all the elements in the set are negative > square roots of prime numbers, how can the size be anything BUT > a negative square root of a prime number? Do you agree with that one. Do you, or do you not, remember the proof? Did I, or did I not, prove that for each natural number there are always only a finite number of predecessors, making an infinite set impossible? > Or what is the size of > { orange, kumquat, banana, apple, grapefruit} carambola > For some sets the size is not a natural number. The obvious > example is the set of all natural numbers. There is absolutely > no reason to believe that the size of a set must be an element > of the set. In the case of the natural numbers there most certainly is. >> Or how the size of >> a set can be simultaneously finite and nonexistent. > Of course, this set size, being equal to the largest finite, has only tenuous > existence. We can be sure it is finite, but we can never identify it, for the > same reason that we can never identify the largest finite natural, and in this > sense it may be considered not to exist, while retaining the property of > finiteness. Every set has a size because it contains element (or none), but in > the case of such a boundless set, that size can never be determined, although > it is known to be finite. > If it does not exist, it is not finite. If unicorns do not exist, they do not have horns. Do unicorns have horns? >> You never shared your opinion on the smallest even prime >> larger than 2? Do you think that number is even? > That definition is self-contradictory, and an obvious attempt to make me look > stupid, so of course I didn't bother responding to it. > Of course it is self-contradictory, just like the largest finite > natural. So why do you insist that the largest finite natural > has properties, but that the smallest even prime larger than 2 > does not? Look, if you want to discuss the smallest even prime greater than two, then you can say things about it: 1. It is even 2. it is prime 3. it is greater than 2 But, the idea is self contradictory to the extent that it doesn't make sense to discuss it. It is not a limit of anything. There is no useful concept there. On the other hand, set theorists seem to like to discuss nothing more than the largest finite natural, so if we're discussing it, while there may be no largest, it is still a finite natural, and if a set of naturals starting at 1 has a largest, then it is the size of the set. > I am not trying to make you look stupid, I am trying to make > you think about what you are saying. You make yourself look > stupid by obviously refusing to think about what you are saying. If you think I am not thinking then you don't know what thinking is. > According to you > 1) the largest finite number is obviously finite, but > we cannot identify it, because that would lead to > contradictions > 2) the smallest even prime larger than 2 is self-contradictory, > and it is therefore not even > This reasoning does not look consistent to me. The largest > finite number is just as self-contradictory as the smallest > even prime larger than 2 as far as I can see. Can you > explain the difference? I think i just did. If you had some context where the smallest even prime greater than 2 had some relationship to something else, it might be worth considering the properties of being prime and even and greater than 2. In this case we are discussing finiteness vs. infinity, so the finiteness of the largest finite natural is relevant, because that number turns out to be the same as the size of the set of all finite naturals. The lack of such a largest finite natural directly indicates the lack of any specific size for the set. In the one case it's relevant though paradoxical, and in the other just pointlessly self-contradictory. > Shall I entertain every > obnoxious quip? Here's my repsonse, as if the question is relevant: No two > primes share any factors, and evens all have a factor of 2, so no prime besides > 2 is even. Duh. > And for every finite n, n+1 is a larger finite number. Therefore > there is no largest finite number, just as there is no prime > beside 2 that is even. Yet you claim that the largest finite > number is finite. So why is the smallest even prime larger > than 2 not even? It is and it's prime too. It's just not interesting or germaine. > Stephen -- Smiles, Tony === Subject: Re: infinity stephen@nomail.com said: >> But there is no largest member, right? Or is there? >> I am never sure if you think there is one or not. > There is no largest finite, since for any finite you can name, there is another > one after it. We all know that. The set has no finite upper bound, and there is > no largest natural number. We all agree on that, I think. We differ in what > kind of significance we attribute to the boundlessness of the naturals. > So if the largest finite does not exist, then nothing, including > the set size, equals the largest finite. > Something can be the same as something else and not exist, as long as the other > thing doesn't exist either. A Flornkney is a purple flying toadstool. They are > the same thing. Neither exists. Yes but it is also true that a Flornkney is not a purple flying toadstool. Actually something can be the same as something else and not exist even if the other thing *does* exist. It is also true that a Flornkney is George Bush. You can say anything you want about something that doesn't exists. However, you cannot say anything meaningful about something that doesn't exist. -William Hughes === Subject: Re: infinity > stephen@nomail.com said: >> stephen@nomail.com said: > But there is no largest member, right? Or is there? > I am never sure if you think there is one or not. >> There is no largest finite, since for any finite you can name, there is another >> one after it. We all know that. The set has no finite upper bound, and there is >> no largest natural number. We all agree on that, I think. We differ in what >> kind of significance we attribute to the boundlessness of the naturals. >> So if the largest finite does not exist, then nothing, including >> the set size, equals the largest finite. > Something can be the same as something else and not exist, as long as the other > thing doesn't exist either. A Flornkney is a purple flying toadstool. They are > the same thing. Neither exists. Now you are just confusing meanings of 'exist'. What you did is describe something that does not physically exist in the real world and gave it a name. That is a lot like the There are not 10^100 stones you can line up and count. But we can give it a name and reason about it, and there are no logical contradictions as a result. Likewise 'purple flying toadstools' do not cause any logical contradictions. They may not conform with our current understanding of the laws of nature, but they are not inherently contradictory. The largest finite however is inherently contradictory, and I cannot give it a name and reason about it, because it logically does not exist. Supposing a largest finite immediately leads to contradictions. But then again, according to you: In my N, N is the maximal element, but also has a successor, which is different from your way of doing things. So logical consistency does not seem to trouble you so much. >> If you want to say that ordered sets without a largest element >> do not have a size then that is reasonable. Claiming that >> the size is something that does not exist is unreasonable. > Well, that is essentially what I am saying. Then stop saying it is finite. If the set size does not exist, attributing properties to it makes no sense. > If there is a size of such a set of > naturals starting from 1, then that is also the largest element, and if there > is a largest element, then that is the set size, but if we prove there cannot > be any largest element, then there cannot be any set size, since that would be > equal to the largest element. The largest element and set size are the same > number, even if neither exists. You only get a contradiction by equating them > if in some case one exists and the other doesn't. Which is what you have been doing. Nobody but you thinks the set size must equal the largest element, and nobody but you thinks the largest finite exists at all, even as an 'unidentifiable' number. > Of course, there will eb some > responses like, Yes that is the case. Aleph_0 is the set size but there is no > largest finite, so you have a contradiction. But, you cannot object using your > conclusion as premise. The definition of aleph_0 has nothing to do with largest elements. You are the only one who insists that set size is inextricably bound together. It is perfectly reasonable to claim that the set of all finite natural numbers does not have a size. People like Han and others opposed to any sort of actual infinity will agree with you. Claiming that aleph_0 means something it does not is not reasonable. It has a precise definition that has nothing to do with largest elements. In a certain sense it also has nothing to do with size, unless you equate cardinality and size. > Remember, you are the one claiming that you can count >> through all the finite naturals at a rate of one per second >> in a finite number of seconds. Nobody else thinks this >> is possible. > That must be because they think there is some finite natural that take an > infinite number of seconds to count to. Which one is that? That is just your quantifier dyslexia talking. >> I beg your pardon? If no element takes an infinite amount of time to count to, >> then what are you counting after an infinite amount of time? You know what you >> can do with your accusations of QD. >> I am not counting after an infinite amount of time. I do >> not know what you even mean by that. If I start counting >> the finite naturals, I will never stop. There is no last >> element. For every finite time, there are numbers I will >> not have reached yet, and at no finite time will I have >> reached every finite number. > And at no infinite time will you still eb counting finite numbers. You will > count for all finite iterations, but no infinite iterations. Yes. So? What does that have to do with your claim that you can count all the finite numbers in a finite amount of time? > There > is no finite natural that takes an infinite number of > seconds to count to. But you can always keep counting. > No matter how many seconds you count, there are still > more finite naturals. If you stop after a finite amount > of time, there will still be an infinite number of finite > naturals you have not yet reached. >> Yes, you count for all finite time, but you do NOT count for an infinite amount >> of time, or you would be counting infinite naturals, not finite ones. >> You need to define precisely what you mean by an infinite amount of time. >> I guess you want people to count forever, and then somemore. >> Do you think there is time after forever? This is one of the >> reasons why bringing time into discussions of infinity seems >> to hinder more than help. > Think of it as times, or iterations. When you count an infinite number of > iterations, you are up to infinite values, one unit of value for each > iteration. That does not clarify things in the slightest. There is no 'after an infinite number of iterations' either. You will never stop going through the finite iterations. The loop for (i=0; i>=0; ++i) cout << i << endl; does not end. It never stops iterating through finite values at i. There is no point at which i goes from finite to infinite. Tony, you are the one claiming that you can count all of the >> finite naturals in a finite amount of time. You seemed to >> have changed your position in mid argument. > No, I am not changing my position. You see an unbounded set and think it's > infinite, but by the properties of the elements, it cannot be. No finite number > is infinite, therefore no number in the set has an infinite number of > predecessors, and the set is finite. Can you identify the end? No. Does that > make it infinite? No. How much time is required to count all the finite natural numbers? You claim that it is finite. So what is the value of the finite number? Oh right, it is probably one of your 'unidentifable finite' numbers. >> So once again, do you think that you can count all of the >> finite naturals in a finite amount of time if you count one >> natural per second? And by count all of them, I mean >> all of them, starting with 1, in one interval of finite time. > Can I tell when it will end? No. Will it go on for an infinite number of > iterations? No. You said it will only take a finite amount of time. >> Note, I do not mean each of them. We all agree that given >> a single finite natural, I can count to that finite natural >> in a finite amount of time (ignoring my own mortality and >> the eventual heat death of the universe). The question is >> can I count >> 1, 2, 3, 4, .... >> and say the name of each and every finite natural number >> in a finite amount of time. > If you have a finite number of them, and they each take finite time, then it > takes finite time. Yes, that is your claim. Therefore, because you claim there are only a finite number of finite numbers, you think that you can count all of finite numbers in a finite number of seconds. What is that number? > You believe you have an infinite number of them, but an > infinite number of increments makes an infinite sum, so you will have infinite > values if you use an infinite number of iterations. You need a finite, but > unbounded number of iterations. My claim is that you cannot count all the finite numbers in a finite amount of time. Pick any finite number k. In k seconds you can only count to k. There are finite numbers larger than k. Therefore you cannot count all the finite numbers in k seconds. k was an arbitrary finite number, so therefore there does not exist a finite amount of time in which you can count all finite numbers. Your 'finite but unbounded' is meaningless when applied to a single value. A single value cannot be 'finite but unbounded' by any reasonable interpretation of 'finite' and 'unbounded'. So far your non-standard approaches so far are not very >> consistent. And insisting on using a word differently >> than its standard meanings is just asking for confusion. > Infinity is the right word. The concept needs fixing. Notice I do NOT use > cardinality. That is a registered trademark of CantorCo. And infinite set is a registered trademark of DedekindCo. You keep using 'infinite set' and meaning something else however. > Infinity is not. I > define it differently, and I am allowed to. Sure, you are also allowed to define apple to mean gorilla, but is not going to help you communicate with anybody. Defining infinity in such a way that it excludes things that never end is just asking for confusion. Of course you have not actually ever defined infinity in a consistent and usable way. > You refuse to share your definition of infinite as it applies > to sets. I saw all that stuff about 0 you never explained how that applied to sets. >> Yes, I defined finite sets as sets with a finite NUMBER of elements, after >> defining finite NUMBER. >> But you did not describe how you determine the NUMBER of elements >> of a set. What is the NUMBER of elements in the set of finite >> naturals? You have been asked this question dozens of times, >> and I have still yet to see a real answer from you. > Then try opening your eyes. I have explained the relationship between this size > and the largest member of this set. But there is no largest member of the set. What good does it do to explain how something is related to something that does not exist? > If you didn't get is the first dozens of > times then I don't know what to do for you. There is no distinct size. Claiming > there is is a mistake. Then there is no size. I can accept that as an answer. However you then cannot claim that there is a size, as you do in every other post. You also have to throw out all your proofs about the size of the set of finite naturals. If the size does not exist, you cannot prove anything else about it. > I asked you > what x was for the finite naturals and you refused to answer. >> What x was? What does that question mean? x is ANY finite natural, in the >> definition I suggested. >> No, you claim that the number of finite naturals is a finite >> number. According to your definition, the number of finite >> naturals equals 1/x where x>0. What is x? Why should >> I believe x is greater than 0, and that 1/x is therefore >> finite according to your definition, if you cannot tell >> me what x is? > Do you understand how definitions work? These are the conditions for > determining if ANY number x is finite: > 1. 0 2. y is finite and > A. x=0-y or > B. x=1/y > Now, if you tell me all n in N are finite, then the set is finite, the size > being the largest finite number. I would never tell you that. And you are supposed to be starting with x, and using that to prove that N is finite. Your definition of finite/infinite starts with values of x, not with sets. > What is the inverse of that? The smallest > positive number, some tiny non-zero indeterminate number. Is that what you want > to know? Do you not understand x is a variable in this definition? Do you think > I can name the reciprocal of a number that can't be named? If you can't name it, then you cannot prove that it is larger than 0. If you cannot prove that it is larger than 0, then you cannot prove that 1/x is finite. Look, if you start with the definitions of finite and infinite values, then your proofs need to use those definitions You are trying to prove that the set of finite numbers is finite. Given your definitions, a proof could look like: 1) the set of finite numbers is finite 2) because the number of finite numbers is finite 3) because the number of finite numbers equals 1/x and 0 If you cannot tell me what x is, how do I know that it is > greater than 0? >> What are you talking about? The largest finite again? I don't understand the >> question. >> See above. > I maintain the set of finites is finite, but that the size cannot be > pinpointed, so certainly the reciprocal of the size also cannot be pinpointed. But you are not using your own definitions in your argument. You produce a definition for finite value, and then promptly ignore it, and just claim without any justification that the set of finites is finite. And you still have not explained how the number of finite > naturals can be a finite natural. >> That I have explained countless times it seems. Remember the proof that all >> sets of naturals starting from 1 have a largest element equal to the set size? >> Yep. >> If all elements in the set are finite naturals, how can the size be anything >> BUT a finite natural? >> That is your argument? If all the elements in the set are negative >> square roots of prime numbers, how can the size be anything BUT >> a negative square root of a prime number? Do you agree with that one. > Do you, or do you not, remember the proof? Did I, or did I not, prove that for > each natural number there are always only a finite number of predecessors, > making an infinite set impossible? The number of predecessors has nothing to do with whether or not a set is infinite, so no, you did not prove anything about infinite sets. >> Or what is the size of >> { orange, kumquat, banana, apple, grapefruit} >> ? > carambola That can't possibly be write, because it is not a member of the set. Your statement: > If all elements in the set are finite naturals, how can the size be anything > BUT a finite natural? clearly implies that you think the size must be a member of the set. >> For some sets the size is not a natural number. The obvious >> example is the set of all natural numbers. There is absolutely >> no reason to believe that the size of a set must be an element >> of the set. > In the case of the natural numbers there most certainly is. Of course those reasons remain safely locked away in your head. If it does not exist, it is not finite. > If unicorns do not exist, they do not have horns. Do unicorns have horns? I have already responded to this misunderstanding of yours. We are not talking about physical existence. Numbers do not physically exist. I have never seen a 2, but it still has a successor named 3, which I also will never see. You never shared your opinion on the smallest even prime > larger than 2? Do you think that number is even? >> That definition is self-contradictory, and an obvious attempt to make me look >> stupid, so of course I didn't bother responding to it. >> Of course it is self-contradictory, just like the largest finite >> natural. So why do you insist that the largest finite natural >> has properties, but that the smallest even prime larger than 2 >> does not? > Look, if you want to discuss the smallest even prime greater than two, then you > can say things about it: > 1. It is even > 2. it is prime > 3. it is greater than 2 > But, the idea is self contradictory to the extent that it doesn't make sense to > discuss it. Just as it is self contradictory to discuss the largest finite. > It is not a limit of anything. There is no useful concept there. On > the other hand, set theorists seem to like to discuss nothing more than the > largest finite natural, so if we're discussing it, while there may be no > largest, it is still a finite natural, and if a set of naturals starting at 1 > has a largest, then it is the size of the set. Set theorists never discuss the largest finite, other than to point out to the people who use it in their arguments that it does not exist. According to you >> 1) the largest finite number is obviously finite, but >> we cannot identify it, because that would lead to >> contradictions >> 2) the smallest even prime larger than 2 is self-contradictory, >> and it is therefore not even >> This reasoning does not look consistent to me. The largest >> finite number is just as self-contradictory as the smallest >> even prime larger than 2 as far as I can see. Can you >> explain the difference? > I think i just did. If you had some context where the smallest even prime > greater than 2 had some relationship to something else, it might be worth > considering the properties of being prime and even and greater than 2. In this > case we are discussing finiteness vs. infinity, so the finiteness of the > largest finite natural is relevant, because that number turns out to be the > same as the size of the set of all finite naturals. The lack of such a largest > finite natural directly indicates the lack of any specific size for the set. In > the one case it's relevant though paradoxical, and in the other just > pointlessly self-contradictory. The largest finite is not relevant to anything. Only you think that it is. Why is a bit of a mystery. You keep invoking it in your proofs, so people keep telling you that it does not exist. You have somehow decided that this means it must be relevant to everyone. I promise you, if you stopped talking about the largest finite, implicitly or explictly, noone else would likely ever mention it again. >> Shall I entertain every >> obnoxious quip? Here's my repsonse, as if the question is relevant: No two >> primes share any factors, and evens all have a factor of 2, so no prime besides >> 2 is even. Duh. >> And for every finite n, n+1 is a larger finite number. Therefore >> there is no largest finite number, just as there is no prime >> beside 2 that is even. Yet you claim that the largest finite >> number is finite. So why is the smallest even prime larger >> than 2 not even? > It is and it's prime too. It's just not interesting or germaine. So you think there exists some x such that 2*x equals the smallest even prime larger than 2? That is the definition of 'even' after all. Stephen === Subject: Re: infinity David R Tribble said: >> Because you do not allow infinite values in the set. It does not stop >> before any finite time, but does not go on for an infinite amount of >> time. This is the Twilight Zone between finite and infinite. > The Twilight Zone is a good place to stick all of TO's TOmatics, as they > certainly don't fit anywhere else. > To be fair, there is indeed a kind of no man's land between finite > quantities and infinite quantities. (But I'm talking here more about > ordinals and cardinals than about naturals.) No successor operation > can produce an infinite quantity from a finite quantity, and no > predecessor operation can produce a finite quantity from an infinite > quantity. > Thus there is, in some sense, an infinitely wide gulf separating the > finites and the infinites. Any finite quantity, no matter how large, > is infinitely smaller than any infinite quantity, no matter how small. > We all agrees with this, but ironically, Tony muddles the concept by > bringing in the unit infinity 'N' and declaring that -N, N/2, log(N), > and so forth are meaningful infinite values. He further muddies the > waters by positing unbounded finite sets, and by requiring infinite > sets to contain infinite values. Well, I am sorry to muddy the pool. Be assured that none of the brown stuff is feces, just the sediment that has been in the pool all along. Underneath it all is a lovely bedrock bottom. So, excuse me while I stir it up and try to uncover the foundation which lies beneath. I would like to point out that, just as there is this gulf between the finite and infinite, similarly there are such gulfs between each of the formulaic N/2 to N, and from N to 2N, and from 2N to N^2, there are infinite number of numbers, and a similar finitely uncrossable zone. That's why one uses infinite units. -- Smiles, Tony === Subject: Re: infinity David Kastrup said: > No it doesn't. As I've said, When we > we? Ain't no one here but us schizophrenics.? > speak of relative sizes of infinite sets, we should do so over some > assumed common value range, or take differing ranges of element > values into account. When we say the size of the set is N, what we > are really saying is that this unit infinity indicates an identity > relation between the values and the positions of the elements in the > set. > Oh, hogwash. > There is one element per unit of element value. If x is the position > of the element in the set, starting from 1, then the value of the > element v(x)=x. That's what the unit infinity really means. It's not > a particular size, but more of a level of density in the real > continuum. > So your opinion is that if I multiply every element of the naturals by > 2 but keep its position in the set, suddenly the set has only half as > many members. Can you name _any_ member that has _vanished_ in the > process of multiplying it by 2? 0 became 0, 1 became 2, 2 became 4, 3 > became 6, and so on. For every member we original had, we got another > member. I already explained this long ago. Maybe you forgot. When you multiply every natural by two, you have also double the difference between any corresponding pair of elements, and therefore doubled the range of values. If you had all naturals up through N, you now have all even naturals up through 2N. You have half the sendit, but twice the range, and therefore maintain the same number of elements in the set. > So if the set has become smaller, where have the missing elements > _gone_? At every _position_ in the set, we still have a unique > value. And if we now divide every element by 2, we have the original > set again. How did the the set suddenly become larger when we just > changed the look of its elements? It didn't. It became more compact in value range, and therefore denser, that is, having more elements per unit of value. If you, instead, take the set of naturals and simply remove all odds, then you have NOT changed the range of values, but you HAVE halved the density, so you DO get a smaller set. -- Smiles, Tony === Subject: Re: infinity > David Kastrup said: No it doesn't. As I've said, When we we? Ain't no one here but us schizophrenics.? speak of relative sizes of infinite sets, we should do so over some > assumed common value range, or take differing ranges of element > values into account. When we say the size of the set is N, what we > are really saying is that this unit infinity indicates an identity > relation between the values and the positions of the elements in the > set. Oh, hogwash. There is one element per unit of element value. If x is the position > of the element in the set, starting from 1, then the value of the > element v(x)=x. That's what the unit infinity really means. It's not > a particular size, but more of a level of density in the real > continuum. So your opinion is that if I multiply every element of the naturals by > 2 but keep its position in the set, suddenly the set has only half as > many members. Can you name _any_ member that has _vanished_ in the > process of multiplying it by 2? 0 became 0, 1 became 2, 2 became 4, 3 > became 6, and so on. For every member we original had, we got another > member. > I already explained this long ago. Maybe you forgot. When you multiply every > natural by two, you have also double the difference between any corresponding > pair of elements, and therefore doubled the range of values. Except that the set of all (finite) naturals cannot have any range of values because it cannot have any maximum differece between values without a specific and concrete largest value, which TO denies exists. > If you had all > naturals up through N, you now have all even naturals up through 2N. You have > half the sendit, but twice the range, and therefore maintain the same number > of > elements in the set. > If you, instead, take the set of > naturals and simply remove all odds, then you have NOT changed the range of > values, but you HAVE halved the density, so you DO get a smaller set. > So that the set of finite evens is half the size of the set of finite evens. The only way that can be is if both sizes are zero or both are infinite. === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > David Kastrup said: >> No it doesn't. As I've said, When we >> we? Ain't no one here but us schizophrenics.? >> speak of relative sizes of infinite sets, we should do so over some >> assumed common value range, or take differing ranges of element >> values into account. When we say the size of the set is N, what we >> are really saying is that this unit infinity indicates an identity >> relation between the values and the positions of the elements in the >> set. >> Oh, hogwash. >> There is one element per unit of element value. If x is the position >> of the element in the set, starting from 1, then the value of the >> element v(x)=x. That's what the unit infinity really means. It's not >> a particular size, but more of a level of density in the real >> continuum. >> So your opinion is that if I multiply every element of the naturals >> by 2 but keep its position in the set, suddenly the set has only >> half as many members. Can you name _any_ member that has >> _vanished_ in the process of multiplying it by 2? 0 became 0, 1 >> became 2, 2 became 4, 3 became 6, and so on. For every member we >> original had, we got another member. > I already explained this long ago. Maybe you forgot. It was proven rubbish last time already. > When you multiply every natural by two, you have also double the > difference between any corresponding pair of elements, Correct. > and therefore doubled the range of values. Wrong, since the range is not given by the difference of any pairs of elements. There is no last finite natural, so you can't talk about its difference with anything. > If you had all naturals up through N, But I did not have all naturals up through some N. I had _all_ naturals, _without_ any limit. > you now have all even naturals up through 2N. Can you name a single finite natural number that is in the new set, but that wasn't in the old set? Of course not. Your consideration only holds for a bounded set. The set of naturals is not bounded. > You have half the sendit, but twice the range, The range is infinite, just like before. There is no number in the new set that has not already been in the old set. None. > and therefore maintain the same number of elements in the set. Name a single number that the new set contains but not the old set. And remember: there is no last finite natural. You agreed to that quite a few times. >> So if the set has become smaller, where have the missing elements >> _gone_? At every _position_ in the set, we still have a unique >> value. And if we now divide every element by 2, we have the >> original set again. How did the the set suddenly become larger >> when we just changed the look of its elements? > It didn't. It became more compact in value range, and therefore denser, that > is, having more elements per unit of value. If you, instead, take the set of > naturals and simply remove all odds, then you have NOT changed the range of > values, but you HAVE halved the density, so you DO get a smaller set. So name a single even finite number that can only be reached by doubling a natural number, but that is not a natural number to start with. Come on, just a single such number. And remember, you can't call the last finite natural number N for such purposes, because there is no last finite natural number. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity <85acie47cx.fsf@lola.goethe.zz> natural by two, you have also double the difference between any corresponding > pair of elements, and therefore doubled the range of values. If you had all > naturals up through N, you now have all even naturals up through 2N. You have > half the sendit, but twice the range, and therefore maintain the same number of > elements in the set. So there are numbers in the doubled set that weren't in the original set? Are there finite numbers in the doubled set that weren't in the original set? - Randy === Subject: Re: infinity >> I already explained this long ago. Maybe you forgot. When you multiply every >> natural by two, you have also double the difference between any corresponding >> pair of elements, and therefore doubled the range of values. If you had all >> naturals up through N, you now have all even naturals up through 2N. You have >> half the sendit, but twice the range, and therefore maintain the same number of >> elements in the set. > So there are numbers in the doubled set that weren't in the > original set? Yes, according to Tony you now have all the even value from N to 2N. Before you only had all the values < N, despite of course the fact that you thought you had *all* the values, regardless of size. Whenever Tony sees the phrase all of he translates it to all less than x for some x. > Are there finite numbers in the doubled set that weren't in the > original set? I do not understand the deep mysteries of TOmatics well enough to even try to answer that one. :) > - Randy === Subject: Re: infinity David R Tribble said: > I think Tony's numbers have a sort of structure like > {0,1,2,3,...} union {...,N-3,N-2,N-1,N} > In that ordering. That is two copies of normal finite natural numbers > pasted next to each other. Though obviously he thinks that these would > be finite sets, so their union must be finite. The second is not finite. It is infinitely larger than the set of finite naturals. > Consider the set G: > G = {N-0, N-1, N-2, N-3, ...}, > where N is Tony's unit infinity 'N' > Would Tony say that 0 is a member of G or not? In my book, the set of all integers, positive and negative, finite and infinite, is one infinite linear set. One can choose any starting point as origin, and define all other numbers as successors and predecessors. So, sure, I would see 0 as a member of a set defined as all successors and predecessors to any whole number. > After all, Tony must think that G is finite, because it has only a > finite number of elements ('N' of them). I thought for a second I was confused, but no, you stated above that N is the unit infinity, so what is this statement supposed to mean? > All of the members of G are > infinite, according to Tony, because 'N' is infinite and so N-k must > be infinite, but this fact is irrelevant to the size of G. If k is always finite and N is infinite, then yes, all n in this set would be infinite. If k is allowed to be infinite, then finite naturals can also be in the set, as well as negatives and negative infinites. > But Tony > would say that there are only N members in G (although we may have to > remove element N-0 to make this true, I'm not sure). And therefore > the last element (which in this case, is also the least element) of G > would be N-N, or 0. I am not sure which N you are talking about. It seems to change every sentence. Like I said, if k is always finite and N is infinite, then you have a finite set with all infinite values. If N is some fictitious largest finite, and you count through all the finite k's, then one could imagine it would end somehow at 0, not that there is a specific number of finites, but whatever that number is, you started there and counted down that many times. > So the size of G is 'N', and G is a finite set containing both a > largest and a smallest element. So N must be a finite, not an > infinite, value like we originally assumed. I don't understand your conclusion, but that's probably because I got lost in the various N's you discussed above. -- Smiles, Tony === Subject: Re: infinity > David R Tribble said: I think Tony's numbers have a sort of structure like {0,1,2,3,...} union {...,N-3,N-2,N-1,N} In that ordering. That is two copies of normal finite natural > numbers pasted next to each other. Though obviously he thinks > that these would be finite sets, so their union must be finite. > The second is not finite. It is infinitely larger than the set of > finite naturals. But has the same cardinality, which is the only useful measure of such sizes. Consider the set G: > G = {N-0, N-1, N-2, N-3, ...}, where N is Tony's unit infinity > 'N' Would Tony say that 0 is a member of G or not? > In my book, the set of all integers, positive and negative, finite > and infinite, is one infinite linear set. Fortunately, TO's book is, and will forever remain, unpublished. > After all, Tony must think that G is finite, because it has only a > finite number of elements ('N' of them). > I thought for a second I was confused, but no, you stated above that > N is the unit infinity, so what is this statement supposed to mean? Since TO calls infinite sets finite, HE is the one who has to come up with a meaning. > I am not sure which N you are talking about. It seems to change every > sentence. That does not even make us even because when TO is talking about N, it can change meanings several times within a sentence. > If N is some > fictitious largest finite, Since TO (and only TO) insists that there is a largest finite which is certainly ambiguous, if not fictitious, it is TO's largest finite, no one else's. > and you count through all the finite k's, > then one could imagine it would end somehow at 0, not that there is a > specific number of finites, but whatever that number is, you started > there and counted down that many times. See what we mean? > I don't understand your conclusion, but that's probably because I got > lost in the various N's you discussed above. It is TO's conclusion, based purely on TO's assumptions, that TO does not seem to understand. But that's fair, as no one else understands, or believes in, TO's conclusions either. === Subject: Re: infinity David R Tribble said: > Stephen said: >> You refuse to share your definition of infinite as it applies >> to sets. I saw all that stuff about 0> you never explained how that applied to sets. > Yes, I defined finite sets as sets with a finite NUMBER of elements, > after defining finite NUMBER. > You didn't come out and say it, but you're implying that you use those > finite numbers you defined to count the elements of sets. How exactly > is this done? More to the point, if there are only a finite number of > finite numbers, and you're using those numbers to count elements, how > do you ever know if a set is infinite? Do you just run out of numbers > before you run out of elements? How is this defined? An infinite is larger than any finite. Basically, it means it doesn't end in a finite number of steps. Now, recursive definitions like the Peano axioms generally describe infinite sets, which do not end. However, the properties of finite numbers are such that, given your restriction of finiteness on the natural numbers, there cannot be more than a finite number of them. You cannot fit an infinite number of them into any finite value range because oo elements *1 unit of value per element equals oo units of value range. If the size is strictly larger than any finite, then it is infinite. If it is always equal to a finite, then it is finite. > What if a set has, say, 'N' elements? How do you count to 'N', which > is infinite, if you don't have enough counting numbers to reach it? > If a set has only 'N'-1 elements, how do you know it's smaller than > a set with 'N'+1 elements if you don't have enough numbers to go > around? I have already suggested the need for infinite naturals in the infinite set. You don't need to count until you run out of finites. That's impossible. You need to apply axioms regarding finiteness and the properties of quantities, symbolic systems, trees, Turing machines, or whatever recursive infinite structure you are examining. I have provided one such important formula regarding symbolic systems, N=S^L, as well ad the inverse function rule for quantitative systems. -- Smiles, Tony === Subject: Re: infinity > David R Tribble said: > Stephen said: >> You refuse to share your definition of infinite as it applies to >> sets. I saw all that stuff about 0> explained how that applied to sets. Yes, I defined finite sets as sets with a finite NUMBER of > elements, after defining finite NUMBER. But TO's definition of finite number depends on prior definition of the algebraic structure of at least the non-negative rationals, which cannot be done, at least by TO, without a priori knowledge of finiteness versus infiniteness. Thus TO's definition is necessarily circular. > I have already suggested the need for infinite naturals in the > infinite set. TO has suggested all sorts of idiocies, but has not been able to overcome the self-contradictions that they create. > You don't need to count until you run out of finites. > That's impossible. If the number of finite naturals were actually finite, then running out of finites would be inevitable, rather than impossible, when trying to count an infinite set. Extract digit, TO. > You need to apply axioms regarding finiteness You need definitions regarding finiteness, not axioms. > I have provided one such important formula regarding symbolic > systems, N=S^L, as well ad the inverse function rule for quantitative > systems. But TO's claim that the set of all finite strings must be finite has a major flaw in that any finite set of finite strings must exclude some finite strings, i.e., any concatentation of all its members, since any such will be longer than every member. TO has not been able to overcome that flaw, so that he studiously ignores all posts that point it out. Proof by willful blindness does not convince mathematicians of anything but your willful blindness. === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > An infinite is larger than any finite. What does larger mean for you with regard to set size? > Basically, it means it doesn't end in a finite number of steps. Sets don't have steps. > Now, recursive definitions like the Peano axioms generally describe > infinite sets, which do not end. Not at all. I can define the following: p(0) = 0 If n is even, then p(n) = p(n/2). If n is odd, then p(n) = p((n-1)/2)+1 That's a recursive definition of a function, and it certainly terminates for n being any natural. > However, the properties of finite numbers are such that, given your > restriction of finiteness on the natural numbers, there cannot be > more than a finite number of them. Whining does not make it so. If there is just a finite number of natural numbers, you can order them and there is a last one at a finite position. And then you could add 1 to it, making it not the last natural number after all. > You cannot fit an infinite number of them into any finite value > range Quite so. But they cover an infinite range without there being any natural at an infinite position. > I have already suggested the need for infinite naturals in the > infinite set. There is no need for anything like that since sets don't need to contain their size as an element. > You don't need to count until you run out of finites. That's > impossible. Right, and therefore there is no last finite natural. But the set size of the finite naturals can't be given by any finite natural. > You need to apply axioms regarding finiteness and the properties of > quantities, symbolic systems, trees, Turing machines, or whatever > recursive infinite structure you are examining. I have provided one > such important formula regarding symbolic systems, N=S^L, as well ad > the inverse function rule for quantitative systems. You have always confused arbitrarily large as a (collective) element property (in the context of the set) with infinite as an element property of an individual element. Quantifier dyslexia. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity David R Tribble said: > Stephen said: >> And you still have not explained how the number of finite >> naturals can be a finite natural. > If all elements in the set are finite naturals, how can the size be > anything BUT a finite natural? > Perhaps because a property of a set is completely divorced from any > property of its elements? So perhaps the set size is not constrained > to be a finite natural, even though all its members are so constrained? Actually, the properties of a set are directly related to the properties of the elements, since a set is nothing else than a number of elements considered as a unit. The only universal property of sets is size, but comparing size for infinite sets also depends on properties of the elements. So, I disagree entirely that properties of elements are divorced from properties of sets. In the case of the natural numbers, they are intimately intertwined. > Consider the set: > B = { > {0,1}, {1,2}, {2,3}, ..., > {0,2}, {1,3}, {2,4}, ..., > {0,3}, {1,4}, {2,5}, ..., > ... > } > In other words, set B contains all possible pairs of natural numbers. > Your question is equivalent to asking: > If all elements in the set are pairs of finite natural numbers, how > can the size be anything BUT a pair of finite natural numbers? Yeah sure, if you provide an inductive proof that shows the set size is always some pair of numbers, then that would be a valid question. > You might also consider how to answer the question, how big is B? N(N+1)/2. Count the size of each column: N, N-1, N-2........2, 1. This is the same half-square as the list of all naturals in unary, and represents the same number as the sum of naturals. > Gee, isn't this fun? Wheeee!!! -- Smiles, Tony === Subject: Re: infinity > David R Tribble said: > Your question is equivalent to asking: > If all elements in the set are pairs of finite natural numbers, how > can the size be anything BUT a pair of finite natural numbers? > Yeah sure Elegant rebuttal! === Subject: Re: infinity Tony Orlow says... >David R Tribble said: >> Your question is equivalent to asking: >> If all elements in the set are pairs of finite natural numbers, how >> can the size be anything BUT a pair of finite natural numbers? >Yeah sure, if you provide an inductive proof that shows the set size is always >some pair of numbers, then that would be a valid question. But you *haven't* proved any of your claims by induction. As I said in another post, the pattern with your proofs by induction are like this: 1. You make claim A, which you prove by induction. Everybody agrees with A. 2. You claim that A implies B. Everybody *disagrees* with that implication. A does *not* imply B. 3. You claim that therefore, B. When people object to B, you say I proved it by induction. But you didn't. You proved A. For example, A might be Forall sets A_n of the form {1, 2, ..., n}, A_n has size n, and n is the last element of A_n. B might be the statement: If U is any set of consecutive naturals whose smallest element is n, then if U has a size, then its size is equal to its largest element Everybody agrees with A. You proved it by induction. But people *don't* agree that A implies B. So it is *false* to say that you proved B by induction. -- Daryl McCullough Ithaca, NY === Subject: Re: infinity David R Tribble said: >> But remind us again what 'N' is? It's not the size of the set of >> naturals, is it? > It's not an exact number. there is no exact size of the set. Given any value > range or set size N, the sum is N(N+1)/2. It does not matter whether N is > finite or infinite, this formula still holds. If we talk about an infinite > range, then we get an infinite value. N as an infinite unit refers to the > identity function which describes it as a set with 1-1 correspondence between > elements and units of element value. N has 1 element per unit of value. The > range may vary. If we say there are N integers, then we may say there are N/2 > naturals. If we say there are N naturals, we may say there are 2N integers. > It's more like a variable than an exact number, but when choosing units, the > choice is rather arbitrary. Sorry if that part's confusing, but I see no > other way that makes any real sense. Well you've been misleading us all this time, then, in asserting that 'N' is the unit infinity. What you really mean is that 'N' is some relative arbitrary unit infinity, which is assigned an arbitrary infinite value for the duration of a set of equations at the moment. The next set of equations that use 'N' will assign it a different infinite value. That being the case, now that we know what 'N' is, or actually what it is not, can you tell us what these specific infinite values might be?: size(Nt) = size of the set of naturals, including infinite naturals size([0,1)) = size of the set of reals in [0,1) size(R) = size of the set of reals, (-oo,+oo) You could also tell us what is the size of the set of finite naturals (the set everyone else calls N), which you say must be a finite value: size(N) = size of the set of finite naturals But please, the continued use of 'N' is only confusing. === Subject: Re: infinity Daryl McCullough said: >If there are no elements in an ordered set with an infinite number of >predecessors, with which you agree, then there are not an infinite number of >elements in the set, since that would imply that some of the elements had an >infinite number of predecessors. > What people are disagreeing with is the *implication* > the set is infinite > - some element has an infinite number of predecessors > Are you claiming to have proved this implication, or are > you claiming that it is an axiom? > -- > Daryl McCullough > Ithaca, NY I am claiming that it is trivially true. For each successor you have a predecessor. If x is the nth successor to y, y is the nth predecessor to x. You cannot have an element in the set with n successors, and no element in the set with n predecessors. No element in the set of finite naturals has an infinite number of successors in the set, or predecessors. If any element had an infinite number of suiccessors in the set, then one of those infinitely far away would have an infinite number of predecessors, each of which was a successor of the first. It's just simply trivially true. -- Smiles, Tony === Subject: Re: infinity > Daryl McCullough said: If there are no elements in an ordered set with an infinite number of >predecessors, with which you agree, then there are not an infinite number >of >elements in the set, since that would imply that some of the elements had >an >infinite number of predecessors. What people are disagreeing with is the *implication* the set is infinite > - some element has an infinite number of predecessors Are you claiming to have proved this implication, or are > you claiming that it is an axiom? -- > Daryl McCullough > Ithaca, NY > I am claiming that it is trivially true. Only in TOmatics. In standard mathematics it is trivially false, any Peano set being sufficient counterexample. TO conflates the impossible (one natural with infinitely many predecessors) with the actual (infinitely many naturals each with a different number of predecessors than any other). === Subject: Re: infinity Tony Orlow says... >Daryl McCullough said: >> What people are disagreeing with is the *implication* >> the set is infinite >> -> some element has an infinite number of predecessors >> Are you claiming to have proved this implication, or are >> you claiming that it is an axiom? >I am claiming that it is trivially true. Once again I ask: does it being trivially true mean that it is an axiom, or that it is trivially *provable* from more basic axioms? >For each successor you have a predecessor. >If x is the nth successor to y, y is the nth predecessor to x. Okay, let that be statement A: A: Forall x, forall y, if x is the nth successor to y, then y is the nth predecessor to x. >You cannot have an element in the set with n successors, and no >element in the set with n predecessors. Okay, let that be statement B: B: forall x, there exists y such that the number of successors of x = the number of predecessors of y How do you prove that A implies B? We agree with A, we disagree with B, and we disagree that A implies B. >If any element had an infinite number of suiccessors in >the set, then one of those infinitely far >away would have an infinite number of predecessors, >each of which was a successor of the first. Once again, let A be the statement: x has infinitely many successors Let B be the statement There exists a y such that y is a successor of x and y has infinitely many predecessors Why do you think that A implies B? >It's just simply trivially true. Is it trivially true because it is an axiom, or is it trivially true because it is provable from more basic axioms? -- Daryl McCullough Ithaca, NY === Subject: Re: infinity > Daryl McCullough said: >>If there are no elements in an ordered set with an infinite number of >>predecessors, with which you agree, then there are not an infinite number of >>elements in the set, since that would imply that some of the elements had an >>infinite number of predecessors. >> What people are disagreeing with is the *implication* >> the set is infinite >> -> some element has an infinite number of predecessors >> Are you claiming to have proved this implication, or are >> you claiming that it is an axiom? >> -- >> Daryl McCullough >> Ithaca, NY > I am claiming that it is trivially true. For each successor you have a > predecessor. If x is the nth successor to y, y is the nth predecessor to x. You > cannot have an element in the set with n successors, and no element in the set > with n predecessors. No element in the set of finite naturals has an infinite > number of successors in the set, or predecessors. If any element had an > infinite number of suiccessors in the set, then one of those infinitely far > away would have an infinite number of predecessors, each of which was a > successor of the first. It's just simply trivially true. It is trivially false. You if x is the nth successor to y, y is the nth predecessor to x only applies to finite n. It is meaningless to talk about the infinityth successor. That is equivalent to assuming that infinity is a natural number, which it is not. 2 has an infinte number of successors. The list of 2's successors never ends. The number of 2's successors is larger than any finite number. The latter is very easy to prove. Let k be any finite number. Consider the set { 3, 4, ...., k, k+1, k+2, k+3 } Because k is finite, all the elements in the set are finite, all the elements are successors of 2, and there are k+1>k elements in the set. For reach and every finite k, it is trivial to prove that 2 has more than k successors. Stephen === Subject: Re: infinity <854q8v364r.fsf@lola.goethe.zz> <85d5ne6ozs.fsf@lola.goethe.zz> What people are disagreeing with is the *implication* > the set is infinite > - some element has an infinite number of predecessors > Are you claiming to have proved this implication, or are > you claiming that it is an axiom? > I am claiming that it is trivially true. For most people that means there exists a short proof, not a foot-stomping declaration that it's obvious, it just HAS to be so!. > For each successor you have a predecessor. Yes. > If x is the nth successor to y, y is the nth predecessor to x. Yes. Also true. So far you're just discussing finite numbers of steps. > You cannot have an element in the set with n successors, and no > element in the set with n predecessors. Yes. > No element in the set of finite naturals has an infinite > number of successors in the set, or predecessors. Incorrect. Every element in the set of finite naturals has an infinite number of successors and a finite number of predecessors. I suspect this is where your QD will jump in again. You perhaps read an infinite number of successors as one successor infinitely far away. There's no particular element which is infinitely far away from x. But there's no end to the number of finite steps I can take from x. Ultimately, this gets back to the question of whether the process of taking finite steps away from x ever stops. If it doesn't, then the process is infinite. Trivially. Finite means the process of taking steps stops. You want it to be unending but still be, in some mysterious sense, finite. > If any element had an > infinite number of suiccessors in the set, then one of those infinitely far > away would have an infinite number of predecessors, Only if one of them was infinitely far away. Every element has an infinite number of finitely-far successors, and no successors which are infinitely far away. - Randy === Subject: Re: infinity <854q8v364r.fsf@lola.goethe.zz> <85d5ne6ozs.fsf@lola.goethe.zz> You cannot have an element in the set with n successors, and no >> element in the set with n predecessors. > Yes. With a caveat: This is true only if n is an honest-to-golly finite natural number (extra caveats to Tony: one of those finite numbers that exist, are distinct, etc., etc.) Certainly if we allow n to be infinity, the statement is false. -- I don't want to wine and dine and date you once or twice. I want to hold you now. I just want to spend the night. You tell me a better plan. Baby, I'm not a patient man. -- Jimmy Lafave, the romantic troubadour. === Subject: Re: infinity Randy Poe said: > If there are no elements in an ordered set with an infinite number of > predecessors, with which you agree, then there are not an infinite number of > elements in the set, since that would imply that some of the elements had an > infinite number of predecessors.. > Why would it imply that? By what principle other than your > demanding it must be true because you want it to be true? > Show me what this follows from other than your own insistence. > Argumentum ad pedem supplosionem [1] is common on sci.math, > especially with you, but it is not a recognized form of valid > deduction. > BTW, I've opened a new thread if you want to talk about limits > and limit notation. > - Randy > [1] pedem supplosionem = stomping of the foot If x is the nth successor of y, isn't y the nth predecessor of x? It seems trivially true that if some element in a set has n predecessors, then another in the set has n successors, and vice versa. Next thing, you'll want mew to prove that every infinite number is larger than every finite, or some other such obvious fact. If you reflect for just a moment, can you not see that one of the infinite successors of an element must have an infinite number of predecessors between it and the first number? In fact, the vast majority would. -- Smiles, Tony === Subject: Re: infinity > Randy Poe said: If there are no elements in an ordered set with an infinite number of > predecessors, with which you agree, then there are not an infinite number > of > elements in the set, since that would imply that some of the elements had > an > infinite number of predecessors.. Why would it imply that? By what principle other than your > demanding it must be true because you want it to be true? > Show me what this follows from other than your own insistence. Argumentum ad pedem supplosionem [1] is common on sci.math, > especially with you, but it is not a recognized form of valid > deduction. BTW, I've opened a new thread if you want to talk about limits > and limit notation. - Randy [1] pedem supplosionem = stomping of the foot > If x is the nth successor of y, isn't y the nth predecessor of x? It seems > trivially true that if some element in a set has n predecessors, then another > in the set has n successors, and vice versa. Relevance? > Next thing, you'll want mew to > prove that every infinite number is larger than every finite, or some other > such obvious fact. If TO makes any claim, he should be able to justify it. So far, TO has made many claims and been able to justify virtually none of them. > If you reflect for just a moment, can you not see that one > of the infinite successors of an element must have an infinite number of > predecessors between it and the first number? Relevance? It is the number of successors, not the number of predecessors of any one of them, that is unbounded. Or does TO have in mind some bound on the number of finite successors than a natural can have? === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > If x is the nth successor of y, isn't y the nth predecessor of x? Sure. > It seems trivially true that if some element in a set has n > predecessors, then another in the set has n successors, and vice > versa. Sure. > Next thing, you'll want mew to prove that every infinite number is > larger than every finite, or some other such obvious fact. There are no infinite numbers in the set of naturals. > If you reflect for just a moment, can you not see that one of the > infinite successors of an element There is no infinite successor of an element in the naturals. There is merely an infinite number of finite successors. Successors exist at arbitrarily large finite distances, but not at infinite distances. You just don't understand the difference between arbitrarily large and infinite. Quantifier dyslexia. There is no largest natural, finite or infinite, there is only an infinite number of finite naturals none of which is the largest or last one. > must have an infinite number of predecessors between it and the > first number? In fact, the vast majority would. There is no infinite successor. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity Jiri Lebl said: > > Do you think I was trying to > > prove that any infinite set must have elements with an infinite number of > > predecessors? That should be obvious. Is it not? > It is not obvious. You have to prove that. > It is interesting to note why tony thinks it's infinite. Tony thinks > that something is infinite if it goes TO infinity as if infinity is > some point. Tony has a hard time imagining a set that doesn't end. > He thinks that any ordered set MUST have two ends. If an ordered set > has both ends, then it is obvious that there have to be elements with > infinitely many predecessors. So it all stems from the problem of Tony > to imagine an UNending ordered set. Even for the finite numbers where > he said there is no largest finite, he still is only halfway in > understanding. Still there is some end, just unidentifiable. > Jiri One can declare an end at infinite distance. Think of [0,1], a unit segment of the real number line: 0.....................................1 It has an infinite number of points in it. If you start at zero, and add one real number at a time, you will never get any finite portion of that interval counted in any finite time. Does this mean the rest of the interval beyond zero doesn't exist? No, it's there, but moving an infinitesimal amount at a time, it will take an infinite number of steps to traverse the finite unit interval. Any finite number of steps leaves you right next to zero, where all the infinitesimals are clustered.. Now, multiply every value by unit oo such that 0->1 and 1->oo: 1.....................................oo It has an infinite number of natural numbers and unit intervals in it. If you start at 1, and add one natural number at a time, you will never get any infinite portion of that interval counted in any finite time. Does this mean the rest of the interval beyond the finites doesn't exist? No, it's there, but moving a finite amount at a time, it will take an infinite number of steps to traverse the entire infinite interval. Any finite number of steps leaves you right next to 1 in this interval, where all the finites are clustered. Now, look at both. Are there numbers beyond the unit interval? Of course. Are there numbers after the infinite interval? Sure. We can make that interval twice as long and say it's 2*oo, just like we can make the unit interval twice as long and say it's 2. Declaring a unit infinity is like declaring any unit. This is how I see the infinite unit. Maybe this clarifies things a bit, although it's not exactly in axiomatic format. I work with pictures more than words when it comes to math. It works 1,000 times better, or so I've heard. -- Smiles, Tony === Subject: Re: infinity > Jiri Lebl said: > > Do you think I was > > trying to > > prove that any infinite set must have elements with an infinite number > > of > > predecessors? That should be obvious. Is it not? It is not obvious. You have to prove that. It is interesting to note why tony thinks it's infinite. Tony thinks > that something is infinite if it goes TO infinity as if infinity is > some point. Tony has a hard time imagining a set that doesn't end. > He thinks that any ordered set MUST have two ends. If an ordered set > has both ends, then it is obvious that there have to be elements with > infinitely many predecessors. So it all stems from the problem of Tony > to imagine an UNending ordered set. Even for the finite numbers where > he said there is no largest finite, he still is only halfway in > understanding. Still there is some end, just unidentifiable. Jiri > One can declare an end at infinite distance. Think of [0,1], a unit segment > of > the real number line: > 0.....................................1 > It has an infinite number of points in it. If you start at zero, and add one > real number at a time, you will never get any finite portion of that interval > counted in any finite time. Does this mean the rest of the interval beyond > zero > doesn't exist? No, it's there, but moving an infinitesimal amount at a time, > it > will take an infinite number of steps to traverse the finite unit interval. > Any > finite number of steps leaves you right next to zero, where all the > infinitesimals are clustered.. > Now, multiply every value by unit oo such that 0->1 and 1->oo: > 1.....................................oo Does TO claim that there is some number , call it x, such that 0*x = 1 and 1*x = oo ? Then TO must have a rule for y -> y*x, for all y with 0 < y < 1. If there is some number x, there must be some rule for multiplication by it. Okay, TO, show us that rule === Subject: Re: infinity > Do you think I was trying to > > prove that any infinite set must have elements with an infinite number of > > predecessors? That should be obvious. Is it not? It is not obvious. You have to prove that. > It is interesting to note why tony thinks it's infinite. Tony thinks > that something is infinite if it goes TO infinity as if infinity is > some point. Tony has a hard time imagining a set that doesn't end. > He thinks that any ordered set MUST have two ends. If an ordered set > has both ends, then it is obvious that there have to be elements with > infinitely many predecessors. So it all stems from the problem of Tony > to imagine an UNending ordered set. Even for the finite numbers where > he said there is no largest finite, he still is only halfway in > understanding. Still there is some end, just unidentifiable. > Jiri > One can declare an end at infinite distance. One can? A curious theme running through your writings, Tony, is the use of the word declare. Consider something that never ends. Declare that the end is at P, where P has sufficient properties of evasiveness that we avoid contradiction. What on earth does this declare mean? Have you ever seen it in a maths book? (BTW, some of your terms look very much like misappropriated programming terms: declaring a unit infinity, like declaring an array of ints.) > ... Think of [0,1], a unit segment of > the real number line: > 0.....................................1 > It has an infinite number of points in it. If you start at zero, and add one > real number at a time,... When you say one real number at a time, in what order? You're not hoping to recite a real number immediately followed by the adjacent real to the right, are you? Because consider any real (e.g. 0.2), then the next adjacent real does not exist. (Not that I suppose that stops you using it.) > ... you will never get any finite portion of that interval > counted in any finite time. Does this mean the rest of the interval beyond zero > doesn't exist? No, it's there, but moving an infinitesimal amount at a time, it > will take an infinite number of steps to traverse the finite unit interval. Any > finite number of steps leaves you right next to zero, where all the > infinitesimals are clustered.. > Now, multiply every value by unit oo such that 0->1 and 1->oo: > 1.....................................oo Uh-huh. Never mind the general problems involved in multiplying by infinity, would you care to give us the tiniest of hints about the mapping you're using here? Perhaps just a third point, after 0 and 1. What does 0.5 map to, or 0.01 ? Brian Chandler http://imaginatorium.org === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > Jiri Lebl said: >> > Do you think I >> > was trying to prove that any infinite set must have elements >> > with an infinite number of predecessors? That should be >> > obvious. Is it not? >> It is not obvious. You have to prove that. >> It is interesting to note why tony thinks it's infinite. Tony >> thinks that something is infinite if it goes TO infinity as if >> infinity is some point. Tony has a hard time imagining a set that >> doesn't end. He thinks that any ordered set MUST have two ends. >> If an ordered set has both ends, then it is obvious that there have >> to be elements with infinitely many predecessors. So it all stems >> from the problem of Tony to imagine an UNending ordered set. Even >> for the finite numbers where he said there is no largest finite, he >> still is only halfway in understanding. Still there is some end, >> just unidentifiable. > One can declare an end at infinite distance. Think of [0,1], a unit > segment of the real number line: > 0.....................................1 > It has an infinite number of points in it. If you start at zero, and > add one real number at a time, you will never get any finite portion > of that interval counted in any finite time. Does this mean the rest > of the interval beyond zero doesn't exist? No, it's there, but > moving an infinitesimal amount at a time, it will take an infinite > number of steps to traverse the finite unit interval. Any finite > number of steps leaves you right next to zero, where all the > infinitesimals are clustered.. While it is a hilarious source of amusement, it does not particularly help your case if you try bolstering your deficiency in one area where you don't have a clue with help from an other area where you are even more clueless. Stick to natural numbers for now. What you write about real numbers above is even more absurd and stupid than what you write about naturals. > Now, multiply every value by unit oo such that 0->1 and 1->oo: > 1.....................................oo > It has an infinite number of natural numbers and unit intervals in > it. If you start at 1, and add one natural number at a time, you > will never get any infinite portion of that interval counted in any > finite time. Does this mean the rest of the interval beyond the > finites doesn't exist? No, it's there, but moving a finite amount at > a time, it will take an infinite number of steps to traverse the > entire infinite interval. Any finite number of steps leaves you > right next to 1 in this interval, where all the finites are > clustered. Complete bull, but you probably knew as much already. > Now, look at both. Are there numbers beyond the unit interval? Of > course. Are there numbers after the infinite interval? Sure. We can > make that interval twice as long and say it's 2*oo, just like we can > make the unit interval twice as long and say it's 2. Declaring a > unit infinity is like declaring any unit. So you don't have any clue about units, either. > This is how I see the infinite unit. Maybe this clarifies things a > bit, although it's not exactly in axiomatic format. I work with > pictures more than words when it comes to math. It works 1,000 > times better, or so I've heard. Just what picture have you heard this from? -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity Dik T. Winter said: > > Dik T. Winter said: > ... > > > For ANY finite length L, there are a finite set of strings of length > > > L or less. For which L in N is the set of all strings up to that > > > length an infinite set? For NONE of them. There is not ONE finite > > > string which, in the ordered set of strings, has an infinite set of > > > predecessors. Therefore the set is finite, since any infinite ordered > > > set MUST have some members (an infinite number) which have an infinite > > > number of predecessors. > > The therefore does not follow, and the since part is what you are trying > > to prove. > > The since part is NOT what I am trying to prove. > Ok, I correct. The since part is equivalent to what you are trying to prove. Not. I am proving here that a set of finite length strings from a finite alphabet is a finite set. The fact that an infinite ordered set will contain elements with an infinite number of predecessors is a fact used in making the argument, but is not equivalent to the argument. It's a general statement about infinite ordered sets. > > Do you think I was trying to > > prove that any infinite set must have elements with an infinite number of > > predecessors? That should be obvious. Is it not? > It is not obvious. You have to prove that. Is it not true that if x is the nth successor to y, that y is the nth predecessor to x? I feel like you want me to prove the sky is blue. Do mathematicians understand anything besides axiomatic proof these days? Do you ever pciture what you're talking about? If you have your set arrayed along a line, and from element x you can take n steps to the right to arrive at y, can you not then take n steps to the left, and arrive back at x? Of course you can, unless elements fell out of the set as you were stepping right, but that's not what happens to sets. This is obvious. -- Smiles, Tony === Subject: Re: infinity > Not. I am proving here that a set of finite length strings from a finite > alphabet is a finite set. For EVERY finite set of finite strings, any concatenation of all those strings is a finite string, and is of necessity longer that any one of them, and thus is again of necessity not in the original set. So that while there are lots of finite sets of finite strings, The st of ALL finite strings cannot be one of them. TO has been faced with this simple and irrevocable proof of his error many times and chooses to ignore it. But it will not go away. So until TO can figure out how to shrink a concaternation of two or more finite strings to the length of one of them, (1) TO will remain wrong and (2)A set of ALL finite strings cannot be finite. > The fact that an infinite ordered set will contain elements with an > infinite number of predecessors is a fact used in making the > argument, but is not equivalent to the argument. It's a general > statement about infinite ordered sets. In the first place, outside of TOmatics, what TO claims to be a fact is NOT a fact. His general statement is particularly wrong for the set of naturals. > Do you think I was trying to prove that any infinite set must > > have elements with an infinite number of predecessors? That > > should be obvious. Is it not? It is so unobvious that it is false. A strictly increasing infinite sequence, considered as a set of values, contains infinitely many values, but each value has only finitely many predecessors in the sequence. I feel like you want me to prove the sky is blue. Considering TO's incapacity with logic, I doubt he could do it. But what TO believes is not evidence, and what he has been able to prove so far is negligible, and does not support what he claims to believe. === Subject: Re: infinity Tony Orlow says... >Is it not true that if x is the nth successor to y, that y is the nth >predecessor to x? Sure, it's true, but that isn't the disagreement. The disagreement is whether x has infinitely many successors implies x has infinitely many predecessors The usual pattern of arguments with you is this: You say one true thing, A, then you say one false thing, B, and then you say that A implies B. What people disagree with is the *implication* A implies B. They believe that A is true, and they believe that B is false, and they believe that (A implies B) is false. Examples: A = the set A_n = {1,2,...,n} is a finite set B = the collection of all finite naturals is a finite set. People don't believe B, and they also don't believe A implies B. (And they also don't believe that ~B implies ~A). But people *agree* with A. A = every finite natural has finitely many predecessors that are finite naturals. B = every finite natural number has finitely many successors that are finite naturals. People don't believe B, and they also don't believe A implies B. (And they also don't believe that ~B implies ~A). But people *agree* with A. A = the set C_n = the set of all character strings of length n or less is a finite set B = the collection of all finite character strings is a finite set. People don't believe B, and they also don't believe A implies B. (And they also don't believe that ~B implies ~A). But people *agree* with A. >I feel like you want me to prove the sky is blue. No, we want you to prove in a noncircular way that x has infinitely many successors implies x has infinitely many predecessors >If you have your set arrayed along a line, and from element x you >can take n steps to the right to arrive at y, can you not then take >n steps to the left, and arrive back at x? Of course you can, >unless elements fell out of the set as you were stepping right, >but that's not what happens to sets. This is obvious. Once again, A = if x is the nth successor to y, then y is the nth predecessor to x B = if x has only finitely many finite predecessors, then x has only finitely many finite successors We agree with A, but we don't believe B, and we don't believe A implies B, and we don't believe ~B implies ~A. -- Daryl McCullough Ithaca, NY === Subject: Re: infinity If you have your set arrayed along a line, and from element x you >>can take n steps to the right to arrive at y, can you not then take >>n steps to the left, and arrive back at x? Of course you can, >>unless elements fell out of the set as you were stepping right, >>but that's not what happens to sets. This is obvious. > Once again, > A = if x is the nth successor to y, then y is the nth > predecessor to x > B = if x has only finitely many finite predecessors, then > x has only finitely many finite successors > We agree with A, but we don't believe B, and we don't believe > A implies B, and we don't believe ~B implies ~A. Put differently, A: For every finite n, if there is an x with n successors, then there is a y which is the nth successor of x. B: If there is an x with infinitely many successors, then there is a y which is the infinity'th successor of x. It really is the same old sort of claim all over again. Each time he's asked to prove something like this, he drags up another similarly dubious claim. For instance, here he may well reply that if x has infinitely many successors then the range of the set of successors is infinity and hence there is a successor infinitely far from x. And so it goes. One dubious claim props up a very similar and similarly dubious claim, ad infinitum. Or at least ad non-distinct, unboundedly large, quasi-existent-um. -- Jesse F. Hughes That's the base tautological space where by tautological space I mean a region of truth. -- James S. Harris does philosophy of mathematics. JSH is a renaissance man. === Subject: Re: infinity > Once again, > A = if x is the nth successor to y, then y is the nth > predecessor to x > B = if x has only finitely many finite predecessors, then > x has only finitely many finite successors > We agree with A, but we don't believe B, and we don't believe > A implies B, and we don't believe ~B implies ~A. Tony's argument is the following: if y has an infinite number of successors, then y must have an infinityth successor if x is the infinityth successor to y, then y is the infinityth predecessor to x therefore x has an infinite number of predecessors The fact that infinityth is not defined and does not exist for an infinite set is something I think he really cannot understand. Stephen === Subject: Re: infinity Do you think I was trying to > > prove that any infinite set must have elements with an infinite number of > > predecessors? No, you have never tried to prove that. You just keep asserting it without proof, and that unproven (and false) statement is at the heart of many of your proofs. A proof with an unproven (especially false) statement in it is not a valid proof. > That should be obvious. Is it not? > It is not obvious. You have to prove that. And in fact you can't prove it, as it is not true. > Is it not true that if x is the nth successor to y, that y is the nth > predecessor to x? Yes. What is the leap from that to your claim that an infinite ordered set must have an element with an infinite number of predecessors? > I feel like you want me to prove the sky is blue. No, just this unproven statement about infinite sets. You keep making statements about finite sets, then saying the above MUST be true about infinite sets. It just MUST. It HASTA be. It IS. It IS IT IS IT IS I CAN'T HEAR YOU LALALALALA. But that's not proof either. > Do > mathematicians understand anything besides axiomatic proof > these days? Not in axiomatic systems, such as the Peano model of the naturals. There are things derivable from the axioms (true statements), and things contradictory to the axioms (false statements). There's no class of truth which is I want it to be true so therefore it is even if it raises contradictions. > If you have your set arrayed along a line, and from element x you can take n > steps to the right to arrive at y, can you not then take n steps to the left, > and arrive back at x? Yes. So you've just established that between any two elements a finite distance apart, the distance can be measured in either direction. What does that have to do with the claim that there's an element infinitely far away? > Of course you can, unless elements fell out of the set as > you were stepping right, but that's not what happens to sets. This > is obvious. But nothing to do with the theorem you want to be true about infinite sets and infinite elements. You make a couple of vacuous statements we can all agree on about finite distances, and say nothing about why you believe this thing about infinite distances. - Randy === Subject: Re: infinity > Dik T. Winter said: > ... > > > For ANY finite length L, there are a finite set of strings of length > > > L or less. For which L in N is the set of all strings up to that > > > length an infinite set? For NONE of them. There is not ONE finite > > > string which, in the ordered set of strings, has an infinite set of > > > predecessors. Therefore the set is finite, since any infinite ordered > > > set MUST have some members (an infinite number) which have an infinite > > > number of predecessors. > > The therefore does not follow, and the since part is what you are trying > > to prove. > > The since part is NOT what I am trying to prove. > Ok, I correct. The since part is equivalent to what you are trying to prove. > Not. I am proving here that a set of finite length strings from a finite > alphabet is a finite set. The fact that an infinite ordered set will contain > elements with an infinite number of predecessors is a fact used in making the > argument, but is not equivalent to the argument. It's a general statement about > infinite ordered sets. > > Do you think I was trying to > > prove that any infinite set must have elements with an infinite number of > > predecessors? That should be obvious. Is it not? > It is not obvious. You have to prove that. > Is it not true that if x is the nth successor to y, that y is the nth > predecessor to x? I feel like you want me to prove the sky is blue. Do > mathematicians understand anything besides axiomatic proof these days? Do you > ever pciture what you're talking about? > If you have your set arrayed along a line, and from element x you can take n > steps to the right to arrive at y, can you not then take n steps to the left, > and arrive back at x? Of course you can, unless elements fell out of the set as > you were stepping right, but that's not what happens to sets. This is obvious. The problem is that you are making the assumption that you can only take an infinite number of steps to to the right if there is an element y an infinite number of steps to the right. However, assume that there is no element y an infinite number of steps to the right but the set is unbounded. Guess what, we are arguing about whether an unbounded set of finite numbers is infinite. - William Hughes > -- > Smiles, > Tony === Subject: Re: infinity David R Tribble said: > If an element has infinitely many other elements that are larger than it > in the set, then one of those must have infinitely many elements that are > smaller in the set. > That's obviously not true. There are an infinite number of elements > in N that are larger than 3, but only a finite number of members of > N (three, to be exact) are less than 3. And that's true for every > k in N. That's not what I meant. If you have a set S with an element x and you know that x has n successors. Is there an element with 10 predecessors? Trivially, x is the tenth predecessor of whatever element is its tenth successor. If you know that any element has n successors, then you know that at least one other element has n predecessors. So, if there is some element x an infinite number of steps after some other element y, then y is an infinite number of steps before x. If any element has an infinite number of successors in the set, then some other element must have an infinite number of predecessors. > We know that 0 is the least member of N, so it is less than all other > members in N. But there is no greatest element of N (you said so > yourself), so for every given k in N there are more than any finite > number of members in N greater than k. There is no greatest finite natural, but every finite natural number is the size of the set of naturals up to that point. The size of any set of finite naturals is a finite natural. It is not MORE than any finite natural. It is greater than or EQUAL to any finite natural. You are conflating unboundedness with infinity, which seems to be standard practice, unfortunately. > In essence, set N is closed at one end because 0 is the least member > of N, but N is open at the other end because there is no greatest > member (as you said so yourself). So what is true about one end > or direction of the set is not true about the other. Correct. But, since I have shown that no finite natural has an infinite number of predecessors, and since any set with elements having an infinite number of successors would also contain elements with an infinite number of predecessors, no finite natural has an infinite number of successors either, which I am sure you would agree amkes the set finite. The only point I can see you arguing against is the correspondence between elements with certain number of successors to those with the same number of predecessors. But, the point I made in the last paragraph is beyond a doubt in my mind. > There are infinitely many [elements] that are larger if there is one that > takes an infinite number of increments to get to from the first. If there > is such an element, then it takes equally infinitely many decrements to > get from that second element down to the first. Also, if the first is > finite, and we increment it an infinite number of times, the value of the > second, the result of these increments, is necessarily infinite. > There is no such element in N as you describe. All of the members of > N are finite naturals. That's standard hogwash. > If no natural has an infinite number of predecessors (or successors, as > I have just shown above that the two statements are equivalent), and that > is all that there is in the set, then no element of the set has an > infinite number of predecessors, and no portion of the set is infinite. > That is true, no portion of set N (i.e., any subset composed of the > members between any two chosen elements of N) is infinite. > But the entire set is infinite. What? You are saying that no two elements have an infinite number of elements between them, but there are an infinite number in the set overall? That makes no sense. Could you have a telephone pole every 100 feet for ten miles, and not have any one be farther than 1 mile from any other? Maybe if you place them in a spiral...... -- Smiles, Tony === Subject: Re: infinity David R Tribble said: >> That is true, no portion of set N (i.e., any subset composed of the >> members between any two chosen elements of N) is infinite. >> But the entire set is infinite. > What? You are saying that no two elements have an infinite number of elements > between them, but there are an infinite number in the set overall? Yep. We've been saying that for quite a while now. > That makes no sense. Could you have a telephone pole every 100 feet for > ten miles, and not have any one be farther than 1 mile from any other? You seem to be drawing an analogy between finite sets and infinite sets. How about this: Starting at some first pole, you have another telephone pole every foot for an unbounded distance. Any two particular poles you pick are a finite distance apart. But there is an unbounded number of poles; for any pole you pick, there is always another pole one foot farther away (but still a finite distance) from the first pole. There is no pole that is an infinite distance from any other pole, but there is no end to the number of poles. You can't find a last pole, because there isn't one, and the only poles you can find are all a finite distance from the first pole. === Subject: Re: infinity contain elements with an infinite number of predecessors You keep assuming that you can somehow reach out towards the end of an unbounded set and pick an element that is some infinite distance from the front of the set. But we all agree that an unbounded set has no non-finite element you can identify with this property. So you're basing your assertion above on something you've already agreed is impossible. === Subject: Re: infinity > David R Tribble said: If an element has infinitely many other elements that are larger > than it in the set, then one of those must have infinitely many > elements that are smaller in the set. That's obviously not true. There are an infinite number of > elements in N that are larger than 3, but only a finite number of > members of N (three, to be exact) are less than 3. And that's true > for every k in N. > That's not what I meant. It is what you said! >If you have a set S with an element x and > you know that x has n successors. Is there an element with 10 > predecessors? Trivially, x is the tenth predecessor of whatever > element is its tenth successor. If you know that any element has n > successors, then you know that at least one other element has n > predecessors. So, if there is some element x an infinite number of > steps after some other element y, then y is an infinite number of > steps before x. If any element has an infinite number of successors > in the set, then some other element must have an infinite number of > predecessors. No! It only means that there are infinitely many elements that have it as a, possibly distant, predecessor. TO is looking at things backwards again, as usual. The mapping from each natural to its ultimate predecessor in the naturals is like a function from an infinite set to a one-element set. looked at in reverse, that one element hass infinitely many 'successors'. We know that 0 is the least member of N, so it is less than all > other members in N. But there is no greatest element of N (you > said so yourself), so for every given k in N there are more than > any finite number of members in N greater than k. > There is no greatest finite natural, This is a point on which TO waffles. > but every finite natural number > is the size of the set of naturals up to that point. Non sequitur, That some subsets of the naturals are of a particular form does not mean that all of them are. > It is not MORE than any finite natural. For all n in N, Size(N) >= n+1 > n. Thus there is no n in N for which Size(N) = n. > It is greater than or EQUAL to any finite natural. And has just been shown to b e treater than every natural number! > You are conflating unboundedness with infinity, which seems to be > standard practice, unfortunately. Since unboundedness implies Dedekind infiniteness which implies all standard types of infiniteness, at least in ZFC or NBG, standard practice is justified everywhere except in TOmatics, where nothing is justifiable. In essence, set N is closed at one end because 0 is the least > member of N, but N is open at the other end because there is no > greatest member (as you said so yourself). So what is true about > one end or direction of the set is not true about the other. > Correct. But, since I have shown that no finite natural has an > infinite number of predecessors, and since any set with elements > having an infinite number of successors would also contain elements > with an infinite number of predecessors, WRONG! Each of infinitely many successors of any natural has a different number of natural predecessors, at least outside of TOmatics. > no finite natural has an > infinite number of successors either, Except outside TOmatics, where things still work rationally. > But, the point I made in the last paragraph is beyond a doubt in my > mind. It is beyond doubt wrong evertywhere except in TO's TOmatics, which, fortunately, will never corrupt true mathematics. > There is no such element in N as you describe. All of the members > of N are finite naturals. > That's standard hogwash. Which gets swine a good deal cleaner than TO's self-contradictory hogwash. > If no natural has an infinite number of predecessors (or > successors, as I have just shown above that the two statements > are equivalent), and that is all that there is in the set, then > no element of the set has an infinite number of predecessors, and > no portion of the set is infinite. That is true, no portion of set N (i.e., any subset composed of the > members between any two chosen elements of N) is infinite. But the > entire set is infinite. > What? You are saying that no two elements have an infinite number of > elements between them, but there are an infinite number in the set > overall? Precisely! > That makes no sense. Fortunately for actual mathematics, making sense in TOmatics is not a criterion for making sense in actual mathematics. === Subject: Re: infinity Randy Poe said: > I am not sure what (4) said, by the way, since you > didn't include that part. Yes, I did, and you quoted it. It is the pair of equations > (4a) and (4b). > I assumes there was a (4) before that, since he refers to it as a statement, > not the statements. > There is not. What appears immediately before (4a) and (4b) > is (3). > Look again, and see if you missed something. You may have a > problem reading English. > This appears to be your problem. > At any rate, having demonstrated that we can not agree on > the meaning of the words in this freshman-level text, let > us move on some different material. > Later in the chapter, Thomas proves that the harmonic series > 1 + 1/2 + 1/3 + ... diverges. > If we consider the sum of the first n terms of the series, > s_n = 1 + 1/2 + 1/3 + ... + 1/n, > we see that this represents the sum of the areas of n rectangles > each of which is somewhat greater than the area under the > corresponding portion of the curve y = 1/x. Therefore > s_n is greater than the area under this curve between > x = 1 and x = n+1: > s_n > integ(1,n+1) dx/x = ln(n+1). > Do you have any problem with my noting that he is talking about > an arbitrary finite value of n here? Do you disagree? No, I agree. He is comparing each term in the series to the area under an integratible function over a corresponding range, and noting that in each case the term in the series exceeds the area under the curve at that point, and therefore represents a larger number than the area under the curve. So, when he establishes that the area under the curve over infinite range is an infinite value, then he has proven that the series diverges as well, since it's larger. > By taking n sufficiently large, we can make ln(n+1) as large > as we please: > lim(n->oo)ln(n+1) = +oo > Do you disagree that he is still talking about finite n > and finite values ln(n+1), as large a finite value as you > please? No, at this point he is taking the integral over the range from 1 to infinity and noting that this gives an infinite value. For any FINITE n, the limit is NOT +oo, but some finite number. Do you disagree? Does the value become infinite for any finite n? > I have left no words out. That mathematical statement follows > immediately after the statement about making ln(n+1) as > large as we please. It is EQUIVALENT. Thomas is emphasizing > that by taking n sufficiently large, we can make ln(n+1) > lim(n->oo)ln(n+1) = +oo. No words about n becoming infinite > or ln(n+1) becoming infinite appear here. None. There > is merely the notation, and the words above which explain > the precise meaning of that notation. Which is exactly the same as taking the sum over n->oo, or taking the integral over [1,oo]. What is your point? That mathematicians have a pact never to touch infinity? > Since s_n > ln(n+1), this also means > lim(n->oo) s_n = +oo > Note again that he is talking about s_n and at every point in > the above discussion, the values of n under consideration > were finite. As n goes to infinity, the limit goes to infinity. Since the whole argument is that the sum is infinite, how can you think you are NOT going to oo in this type of problem? > Or do you believe there are diverging series which only > diverge when we add in terms with infinite natural indexes, > but that converge when only finite natural terms are > considered? Well, of course, for each finite n, each term 1/n is also finite, and the sum of a finite number of such finite terms is finite. So, yes, I believe that the sum only achieves infinity with an infinite number of terms, much like the sum of the naturals. > - Randy -- Smiles, Tony === Subject: Re: infinity > Randy Poe said: I am not sure what (4) said, by the way, since you didn't > include that part. Yes, I did, and you quoted it. It is the pair of equations (4a) > and (4b). > I assumes there was a (4) before that, since he refers to it as a > statement, not the statements. There is not. What appears immediately before (4a) and (4b) is (3). Look again, and see if you missed something. You may have a > problem reading English. This appears to be your problem. At any rate, having demonstrated that we can not agree on the > meaning of the words in this freshman-level text, let us move on > some different material. Later in the chapter, Thomas proves that the harmonic series 1 + > 1/2 + 1/3 + ... diverges. If we consider the sum of the first n terms of the series, > s_n = 1 + 1/2 + 1/3 + ... + 1/n, > we see that this represents the sum of the areas of n rectangles > each of which is somewhat greater than the area under the > corresponding portion of the curve y = 1/x. Therefore s_n is > greater than the area under this curve between x = 1 and x = n+1: s_n > integ(1,n+1) dx/x = ln(n+1). Do you have any problem with my noting that he is talking about an > arbitrary finite value of n here? Do you disagree? > No, I agree. He is comparing each term in the series to the area > under an integratible function over a corresponding range, and noting > that in each case the term in the series exceeds the area under the > curve at that point, and therefore represents a larger number than > the area under the curve. So, when he establishes that the area under > the curve over infinite range is an infinite value, then he has > proven that the series diverges as well, since it's larger. By taking n sufficiently large, we can make ln(n+1) as large as we > please: lim(n->oo)ln(n+1) = +oo Do you disagree that he is still talking about finite n and finite > values ln(n+1), as large a finite value as you please? > No, at this point he is taking the integral over the range from 1 to > infinity and noting that this gives an infinite value. For any FINITE > n, the limit is NOT +oo, but some finite number. Do you disagree? > Does the value become infinite for any finite n? I have left no words out. That mathematical statement follows > immediately after the statement about making ln(n+1) as large as > we please. It is EQUIVALENT. Thomas is emphasizing that by taking > n sufficiently large, we can make ln(n+1) as large as we please is > n becoming infinite or ln(n+1) becoming infinite appear here. None. > There is merely the notation, and the words above which explain the > precise meaning of that notation. > Which is exactly the same as taking the sum over n->oo, or taking the > integral over [1,oo]. What is your point? That mathematicians have a > pact never to touch infinity? Since s_n > ln(n+1), this also means > lim(n->oo) s_n = +oo Note again that he is talking about s_n and at every point in the > above discussion, the values of n under consideration were finite. > As n goes to infinity, the limit goes to infinity. Since the whole > argument is that the sum is infinite, how can you think you are NOT > going to oo in this type of problem? Or do you believe there are diverging series which only diverge > when we add in terms with infinite natural indexes, but that > converge when only finite natural terms are considered? > Well, of course, for each finite n, each term 1/n is also finite, and > the sum of a finite number of such finite terms is finite. So, yes, I > believe that the sum only achieves infinity with an infinite number > of terms, much like the sum of the naturals. Definition of diverging upwards for real sequences; the sequence f(n), with domain N and range R diverges upwards if and only if for every positive epsilon, (however large) the set { n : f(n) < epsilon} is finite. There is nothing in that definition requiring anything to be infinite, in fact quite the contrary, it is based on the finiteness of any bounded set of values from the sequence.. And any definition of a sequence diverging upward is logically equivalent to the one above. So that TO has his head where the sun don't shine, again! === Subject: Re: infinity The reading lessons continue. I am attempting to walk TO through the chapter in Thomas Calculus Vol 2. TO insists on translating Thomas into TOmatics. > Randy Poe said: > Later in the chapter, Thomas proves that the harmonic series > 1 + 1/2 + 1/3 + ... diverges. > If we consider the sum of the first n terms of the series, > s_n = 1 + 1/2 + 1/3 + ... + 1/n, > we see that this represents the sum of the areas of n rectangles > each of which is somewhat greater than the area under the > corresponding portion of the curve y = 1/x. Therefore > s_n is greater than the area under this curve between > x = 1 and x = n+1: > s_n > integ(1,n+1) dx/x = ln(n+1). > Do you have any problem with my noting that he is talking about > an arbitrary finite value of n here? Do you disagree? > No, I agree. He is comparing each term in the series to the area under an > integratible function over a corresponding range, and noting that in each case > the term in the series exceeds the area under the curve at that point, and > therefore represents a larger number than the area under the curve. OK so far. > So, when he > establishes that the area under the curve over infinite range is an infinite > value, He never does anything over infinite range. You added that. Let's confine ourselves please to what he does write. > By taking n sufficiently large, What does that mean to you? Do you think n has to be made infinite to be sufficiently large? > we can make ln(n+1) as large > as we please: What does that mean to you? Do you think when he talks about making ln(n+1) as large as we please he means infinite values? - Randy === Subject: Re: infinity Daryl McCullough said: >Daryl McCullough said: > What you have said is the following: (Let U = the set of all > finite naturals) 1. If U has a size, then the size of U is less than or equal > to its largest element. >Equal > 2. U doesn't have a largest element. Therefore, U doesn't have a size. >Correct > Note here: Tony says it is correct that U doesn't have a size. 3. U is finite. > 4. A set is finite if and only if its size is some finite natural. >Correct... > Note here: Tony says it is correct that U does have a size. > -- > Daryl McCullough > Ithaca, NY If the size of the set is the same as the value of the largest finite natural, then its size is some finite natural, but since this number doesn't really exist as far as being identifiable, the same holds true for the set size. Paradoxical, yes, but not as nonsensical as Banach-Tarski spheres. -- Smiles, Tony === Subject: Re: infinity > Daryl McCullough said: Daryl McCullough said: What you have said is the following: (Let U = the set of all > finite naturals) 1. If U has a size, then the size of U is less than or equal > to its largest element. >Equal > 2. U doesn't have a largest element. Therefore, U doesn't have a size. >Correct Note here: Tony says it is correct that U doesn't have a size. 3. U is finite. > 4. A set is finite if and only if its size is some finite > natural. >Correct... Note here: Tony says it is correct that U does have a size. -- > Daryl McCullough > Ithaca, NY > If the size of the set is the same as the value of the largest finite > natural, > then its size is some finite natural, but since this number doesn't really > exist as far as being identifiable, the same holds true for the set size. > Paradoxical, yes, but not as nonsensical as Banach-Tarski spheres. Since U has a perfectly well defined cardinality, but apparently does not have any TOsize, why would anyone in their right mind consider TOsize as relevant to anything? Cardinalities work for all sets, even if not the way that TO seems to want, but TOsizes do not work at all for at least one important set. Moral: Stick with what works, even if TO doesn't like it. === Subject: Re: infinity Tony Orlow says... >If the size of the set is the same as the value of the largest finite >natural, then its size is some finite natural You have said that it is true by definition that every finite set has a size that is equal to some finite natural. >but since this number doesn't really exist as far as being >identifiable, the same holds true for the set >size. It sure seems to me that you are saying: 1. Every finite set has a size that is equal to some finite natural. 2. The set of all finite naturals is a finite set. 3. But the set of all finite naturals does *not* have a size. That seems like a direct contradiction to me. -- Daryl McCullough Ithaca, NY === Subject: Re: infinity If the size of the set is the same as the value of the largest finite >natural, then its size is some finite natural > You have said that it is true by definition that every finite > set has a size that is equal to some finite natural. >but since this number doesn't really exist as far as being >identifiable, the same holds true for the set >size. > It sure seems to me that you are saying: > 1. Every finite set has a size that is equal to some finite natural. > 2. The set of all finite naturals is a finite set. > 3. But the set of all finite naturals does *not* have a size. > That seems like a direct contradiction to me. The problem is that TO claims the size of a set of finite numbers exists but cannot be determined. He also claims that neither a: X exists b: X exists but cannot be determined or a': X does not exist b': X exists but cannot be determined is a contradiction. -William Hughes > -- > Daryl McCullough > Ithaca, NY === Subject: Re: infinity David R Tribble said: > Generally, all set size are natural numbers, in my view, because I include > all whole numbers, finite and infinite, as natural. > I believe you said that you no longer consider the size of the set of > naturals to be the same as the size of the set of reals in [0,1]. > What then is the natural number equal to the size of this second set? > How about the size of R? I can't really specify a natural number for it, but there are an infinite whole number of reals in [0,1] aren't there? -- Smiles, Tony === Subject: Re: infinity > David R Tribble said: Generally, all set size are natural numbers, in my view, because I > include > all whole numbers, finite and infinite, as natural. I believe you said that you no longer consider the size of the set of > naturals to be the same as the size of the set of reals in [0,1]. > What then is the natural number equal to the size of this second set? > How about the size of R? > I can't really specify a natural number for it, but there are an infinite > whole > number of reals in [0,1] aren't there? Perhaps TO's inability is because there is no such natural meeting the natural definition of naturals. Just as there is no natural meeting the natural definition of naturals which represents the set of all naturals, even though there are well defined cardinalities for both the set of reals and the set of naturals. === Subject: Re: infinity Randy Poe said: > Daryl McCullough said: Daryl McCullough said: > What you have said is the following: (Let U = the set of all >> finite naturals) >> 1. If U has a size, then the size of U is less than or equal >> to its largest element. > Equal >> 2. U doesn't have a largest element. >> Therefore, U doesn't have a size. > Correct >> 3. U is finite. >> 4. A set is finite if and only if its size is some finite natural. > Correct, but it is not necessary to know WHICH finite natural in order to know > the set is finite. It is sufficient to know that the size is equal to SOME > finite natural for the set to be finite. > It is not necessary to know WHICH finite natural is the size > of U to conclude that the assumption of the size being > finite leads to contradictions. > If the size of U is a finite natural, it is not necessary to > know WHICH finite natural it is, in order to conclude that > the sum of the elements in U > - exists > - is a finite natural > - and is not in U This is true for any given finite natural. Whatever contradiction that follows from these facts depends on the assumption that the size of this U is the largest finite natural, and that U includes all finite naturals. When you assume you have identified the last element and the size of the set, you get a contradiction because that number can never be pinpointed. > To conclude these things, it is sufficient to assume that > the size of U is SOME finite natural. > I said its size was finite, but not identifiable. You get it this time or > never. I am done repeating this point. > Since you've never defined what identifiable means, nobody > can draw any conclusions from this mysterious property. Identifying something means giving it an identity, like a name, such as the size of U or its last element, which specification in this case causes contradictions. > However, we do know that the sum of a finite set of finite > naturals is a finite natural not in the set. We don't know > the value of this sum if we don't know the number of terms > in the sum. But the finiteness is sufficient to tell us > that the set can not contain all finite naturals. That's right. As soon as you have named the size of the set of finite naturals, or its largest element, you get a contradiction, since the set is unbounded. > Identifiable is not required for this proof. Finite > suffices. Suffices to cause a contradiction? No, the source of the contradiction is the assumption that you have specified a finite upper bound for the set. > - Randy -- Smiles, Tony === Subject: Re: infinity > Randy Poe said: > If the size of U is a finite natural, it is not necessary to > know WHICH finite natural it is, in order to conclude that > the sum of the elements in U > - exists > - is a finite natural > - and is not in U > This is true for any given finite natural. Whatever contradiction that > follows > from these facts depends on the assumption that the size of this U is the > largest finite natural Which it precisely what TO keeps trying to sell us. , and that U includes all finite naturals. When you > assume you have identified the last element and the size of the set, > you get a contradiction because that number can never be pinpointed. It does if the number of naturals is finite, as TO keeps saying. To conclude these things, it is sufficient to assume that > the size of U is SOME finite natural. I said its size was finite, but not identifiable. Saying that it exists is not identifying it. > You get it this time or > never. I am done repeating this point. Promises, promises! Since you've never defined what identifiable means, nobody > can draw any conclusions from this mysterious property. > Identifying something means giving it an identity, like a name, such as the > size of U or its last element, which specification in this case causes > contradictions. So that TO says that there is a largest natural which cannot even be called the largest natural because calling it that would cause contradictions? That assertion in itself causes contradictions. However, we do know that the sum of a finite set of finite > naturals is a finite natural not in the set. We don't know > the value of this sum if we don't know the number of terms > in the sum. But the finiteness is sufficient to tell us > that the set can not contain all finite naturals. > That's right. As soon as you have named the size of the set of finite > naturals, or its largest element, you get a contradiction, since the > set is unbounded. Naming of nonexistent thing causes no problems. 'Pegasus', while naming a non-existent, does not cause the universe to collapse. We haven't named it, TO is the one who has 'named' it. Its very existence, even unnamed, causes contradictions, so it is its existence, not its being named, which is prohibited as contradictory. Identifiable is not required for this proof. Finite > suffices. > Suffices to cause a contradiction? No! It is TO's insistence of the existence of the non-existent that causes the contradiction. No, the source of the contradiction is the > assumption that you have specified a finite upper bound for the set. As Randy has not specified any such thing, if anything he has specified the lack of one, TOmatics, not mathematics, is the source of the contradiction. AS usual, and as always! === Subject: Re: infinity > [ ... as usual ... ] Ah, Jesse, here you are ... I have the pleasure to invite you to a new thread, called: Testable Predictions by HdB My best shot, as promised. Han de Bruijn === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> one cannot divide an infinite set into two finite sets, by omitting a single >> element. > That's a good point. > But what about finite sets? Suppose that we start with N, the set of > finite naturals, which you say has only a finite number of elements. > Now we create subsets of N by systematically omitting one element > from it, one at a time. For convenience, we'll just take out the > least element. So we get these subsets: > N = {0, 1, 2, 3, ...} > S_1 = {1, 2, 3, 4, ...} > S_2 = {2, 3, 4, 5, ...} > S_3 = {3, 4, 5, 6, ...} > ... > S_k = {k, k+1, k+2, k+3, ...} > Since we started with a finite set (N), at what point (for what > value of k) do we run out of finite elements to remove and get an > empty set? At noon. You know you were asking for it. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity <85ek7z1ju4.fsf@lola.goethe.zz> <85y85xzety.fsf@lola.goethe.zz> Since we started with a finite set (N), at what point (for what >> value of k) do we run out of finite elements to remove and get an >> empty set? > At noon. You know you were asking for it. :-) I was predicting that Tony would give an unhelpful answer like, after N elements are removed. Perhaps Tony's 'N' is Noon? === Subject: Re: infinity There is no n with an infinite >> number of predecessors. > But every n in N has infinitely many successsors. To see how far one has to go, it does not help to look only at how far > one has come. > TO is looking backwards when he should be looking forwards. > I think that is an essential part of Tony's intuition about > infinity. In order for a set to be infinite you have to > be able to stop at some point and look back and see an > infinite number of things. This is sort of consistent with > his idea that sets are created by adding things one at a time > an infinite number of times. There has to be some point > you can reach where there are an infinite number of things. > I suppose this is the result of thinking of infinity as > being just like any old normal number. > Given the choice of a unit infinity, in many ways the infinite CAN be treated > similarly to the finite. I think this is an important clue to your thinking. Basically, you have not grasped *at all* the idea of considering the totality of an unending sequence of things, such as the pofnats. Your infinity is something you hope to regard as just another number. Now it seems we have made a lot of progress: you agree the sequence of pofnats never ends, and (equivalently) has no last member; you wish to refer to this by saying that the set of pofnats is unbounded. Fair enough. Aside 1: unbounded is a tricky word. In topological terms it's used to mean something orthogonal to (the normal meaning of) infinite - thus an infinite strip is bounded because it has two edges, while the surface of a sphere is finite, yet unbounded because it has no edges. So I suggest Tony UnBounded, and to make a neat word that looks like an adjective, I'm going to call this property tubby. Aside 2: Terminology in general. No-one (I assert) will *ever* get anywhere with you if they persist in using the I-word. But it doesn't matter: mathematics studies the patterns that arise when we define structures, and look at the consequences of what we have defined. It does not matter a jot what words we use. Do you agree with this? (I notice in the latest round of Zickbabble, Lester once again complained that when mathematicians say something about transcendental numbers, that follows from the definition of transcendental, they are wrong, because really the (intrinsic?) meaning of transcendental is something else entirely. Can you agree - whatever you might think of Lester's thesis - that this claim is nonsense: that a person using words to mean what they have defined them to mean can somehow be wrong because the words mean something else?) OK. First big question: if a set is tubby, is any larger set also tubby? For example, you claim that the pofnats are a finite set, that the Tonats are an infinite set, the posnats are a subset of the Tonats, and infinite [numbers] are bigger than finite [numbers]. So are the Tonats also tubby? I think that if so, one might define 'tubby' formally something like: Def. A set is tubby if it includes elements which can be arranged as an unending sequence. Are you happy with this definition? oo-._.-._.-._. Just to go back slightly to my initial comment. David Kastrup just got it all wrong, claiming that (in Tomatics) multiplying all the naturals by two loses half of them. Let me see if I can do better: Take the unending sequence of pofnats, and multiply every one by two. This gives the same number of values, forming a new unending sequence that fails to end twice as far away as the first one. Whereas, striking out every other pofnat yields half as many values, failing to end at the same distance as the original set failed to end. How'd'I do? Basically, you are not considering the entire unending sequence (of course it's nonsense to claim that different things can fail to end at different places, since unending means that there isn't a place that could be somewhere different); rather you are imagining that you can call an arbitrary (not to mention indistinct!) stopping point infinity. Of course, given freedom of terminology, you can; but it's not entirely surprising that it turns out you handle this infinity you refer to in just the same way as a (normal mathematical) finite number. Hmm, this could run and run. Brian Chandler http://imaginatorium.org === Subject: Re: infinity > Virgil said: Jesse F. Hughes said: >> There is no finite n for which there are an infinite number >> of predecessors. Correct. But there also is no finite n for which there is > not an infinite number of successors. I wonder if there are an infinite number of finite integers > (positive and negative whole numbers). Every integer has an > infinite number of predecessors, so at least that argument of > Tony's can't go through. Only if you include infinite negative integers. Then which finite negative integer is one added to some infinite > negative integer? TO has this short circuit in his brain that > prevents him from seeing anything he doesn't want to see. > Largest Negative Finite!! .....mmmmmmmO !hayuh hayuH (snore) TO's childish response to facts he does not want to face, convinces no one of anything but his childishness. There are the same number of finite positive integers as finite > negative integers, and if that number is finite, then twise that > number is finite. But if either is infinite, as is actually the case, they both are, > as is actually the case. > Actually not. Oh, wait. I see it now. There are twice as many integers as > finite numbers, so the size of Z is twice the size of N[1]. > Since N is finite, so is Z. Which might be rather more plausible if he didn't also deny > that the set N has a size. > The set N has no distinct size. We know it is a finite natural, > but it would be the largest finite natural, which doesn't exist, > despite its finiteness. Anyway, I think I'm getting the hang of things. The size of > the set of negative numbers is -1, I betcha. Don't be stupid. Does TO claim proprietary rights on being stupid? Does he have a > patent on it? > Yes, and you owe me about 10 million dollars in fees for your > unauthorized use of my product over the years. Fortunately for me, real courts of law do not operate on TOlogic, so that allegation of debt can only exist in TO's imagination But we should be charging TO tuition in logic and mathematics for our efforts to educate him out of his ignorance. And tuition is collectable even if the student, as in TO's case, learns nothing. === Subject: Re: infinity >The size of ANY set of finite naturals is a finite natural, and any >>initial segment of size n has last element n, > Look, Tony: there it is. You are saying that the size of any set > of finite naturals is a finite natural. >> 1. The set of all finite naturals is a finite set. >> 2. The size of any set of finite naturals is a finite natural. >> 3. Any initial segment of size n has last element n. >No, I said for any n in N there is an initial segment which is finite. >>The size of ANY set of finite naturals is a finite natural, and any >>initial segment of size n has last element n, > You have started to contradict yourself within the same posting. > That's getting bad, Tony. > -- > Daryl McCullough > Ithaca, NY > I am trying to word things carefully, because every statement I make gets > technical objections. I am not sure why I objected to 3, if I did. I am trying > to stick to exactly what the proof says, since it seems have been deemed > acceptable, but still not proving the case for the entire set, based on more > technical objections which really don't make sense. The best way to say it, I > think, is my rephrasing above. If things aren't phrased exactly, people make > all sorts of claims about what I am saying. Well how about adressing the actual question? Here, I will make it even simpler for you. You have stated The size of ANY set of finite naturals is a finite natural. (using exactly those words and the emphasis) As usual let the set of all finite naturals be N. So, the size of N is a finite natural. However, it is easy to see that for every n in N, n is not the size of N, so the size of N is not a finite natural. Can we say the size of N does not exist, but if it did exist it would be a finite natural? Yes, but if the size of N does not exist the the statements -If the size of N exists, the size of N is finite -If the size of N exists, the size of N is infinite -If the size of N exists, the size of N is a pink flying elephant are all true, so we haven't said much. You cannot say anything meaningful about the properties of something that does not exist. - William Hughes === Subject: Re: infinity > David R Tribble said: >> Okay, then what is the sum of all natural numbers? It's an infinite sum. > [...] > No it is not [meaningless], and it's perfectly demonstrable that that is > the answer. Observe the natural numbers in unary: > [...] > Therefore the sum is the sum of (N^2+N)/2 1's. Yes, the triangle number T(k) = k(k+1)/2. But remind us again what 'N' is? It's not the size of the set of > naturals, is it? Because you keep saying that set N has no largest member, and thus > no well-defined size, yet here you are using 'N' like a natural > number that exists and appears to be the size of the set of naturals. > It's confusing, and it seems to be inconsistent with the other things > you've said. > It's not an exact number. There is no such thing as an inexact number. There are unknown numbers, but not inexact ones. > there is no exact size of the set. Every set has an exact size, if size is to mean anything. Dedkind and Cantor have exact sizes for every set, so that TO must be incompetent not to have them. > Given any value > range or set size N, the sum is N(N+1)/2. Since range and set size are not defined for most interesting sets, what TO is claiming is meaningless garbage. > If we talk about an infinite > range, then we get an infinite value. What is this we crap? Only TO talks of ranges, particularly for sets that don't have them. For sets which are metric spaces there are diameters, non-metric sets do not even have diameters, much less ranges. > N as an infinite unit is a contradiction in terms === Subject: Re: infinity <85ek7z1ju4.fsf@lola.goethe.zz> set is ordered, every element is a successor or predecessor (not > immediate) of every other element in the set. Therefore, each > element either has an infinite number of predecessors, or an > infinite number of successors, or both, since all other elements are > either predecessor or successor to that element, and one cannot > divide an infinite set into two finite sets, by omitting a single > element. This fascinating definition of infinite set (in terms of infinite numbers of elements) fails to prove your claim that in an infinite set, some element has an infinite number of predecessors. You have proved that every element has an infinite number of predecessors or an infinite number of successors or both. But the set N satisfies this claim. Every element of N has an infinite number of successors. Yes, I know you deny this, but the reason you give for denying it is that no element has an infinite number of predecessors. But this number must have an infinite number of predecessors. So, if you want to deny that N is infinite, you need a different argument that 0 doesn't have an infinite number of successors. -- Jesse F. Hughes [I]f gravel cannot make itself into an animal in a year, how could it do it in a million years? The animal would be dead before it got alive. --The Creation Evolution Encyclopedia === Subject: Re: infinity <853boabg00.fsf@lola.goethe.zz> <87br2ypfjz.fsf@phiwumbda.org> <871x3s3l3k.fsf@phiwumbda.org> > Surely you can't really claim that the size of Z is twice the size of >> N, since there is no such thing as the size of N. I don't see how you >> can take multiples of something that doesn't exist. > You can note that you have a mirror image of N, in addition to the > original N, which makes two times whatever N is. Despite the fact > that we can't pinpoint N, we can say Z has twice the range of N. I don't know what twice a non-existent quantity is. Oh yeah. I remember now. It's twice an existing but non-distinct quantity. That's okay, then. -- So how do you go on? [...] How will you keep moving for the next few weeks or months until you are known for what you are, the story becomes huge all over the world, and you have reporters at your schools asking you, why? -- Another JSH mystery === Subject: Re: infinity <853boabg00.fsf@lola.goethe.zz> <87br2ypfjz.fsf@phiwumbda.org> <871x3s3l3k.fsf@phiwumbda.org> That's remarkable reasoning, that is. N has a size, but it's not >> distinct, because if it was, it would be something that doesn't >> exist, but it does, so it's not distinct. > Is there a valid objection in there, because I don't see any. No. No objection at all. Aside from not knowing what you mean by distinct of course, which seems to be existing but magically thwarting contradictions. -- A set having three members is a single thing wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as 'three in one' should be child's play. --Max Black, _Caveats and Critiques_ === Subject: Re: infinity > stephen@nomail.com said: >> Daryl McCullough said: The size of ANY set of finite naturals is a finite natural, and any initial >segment of size n has last element n, so if you declared your set to have size >aleph_0, and to be all naturals starting from 1, then you ARE declaring aleph_0 >to be the largest element of the set, and yet there is no such thing. There is >no largest finite, nor smallest infinite, despite your theory. We have statement 1 from Tony: 1. The set of all finite naturals is a finite set. > 2. The size of any set of finite naturals is a finite natural. > 3. Any initial segment of size n has last element n. >> I never said 3. So you are being disingenuous, but that seem to be the norm >> here. >> Yes you did. You even quoted it in this message, 11 lines above >> the line in which you deny saying it. I will cut and paste it >> from above. >>The size of ANY set of finite naturals is a finite natural, and any initial >>segment of size n has last element n, >> The evidence is sitting right there in the message you >> yourself posted. Nobody other than you is being disingenuous. >> Stephen > I am not sure why I denied 3. Is that exactly what it said? Hmmm... I must have > misread or something. I retract the denial, since I don't know what my > objection was. I think i wanted to state it as the proof did. Sorry. Well you did say it Tony. Are you going to now go back and explain what was wrong with the argument? It looks pretty conclusive that your statements logically lead to the conclusion that there is a largest finite number. Stephen === Subject: Re: infinity > stephen@nomail.com said: > Daryl McCullough said: >>The size of ANY set of finite naturals is a finite natural, Claimed without proof, and demonstrably false, except in TOmatics. >>There is no largest finite, nor smallest infinite, despite your >>theory. There is certainly no largest finite nor any infinite in the set of standard naturals, regardless of what idiocies TO may dream up in TOmatics. But that does not limit all numbers to being naturals. And there is a smallest 'number' larger than all naturals which is the 'number' of naturals in the Dedekind-Cantor sense, i.e., cardinality. >> We have statement 1 from Tony: >> 1. The set of all finite naturals is a finite set. >> 2. The size of any set of finite naturals is a finite >> natural. >> 3. Any initial segment of size n has last element n. > I never said 3. So you are being disingenuous, but that seem to > be the norm here. TO is not accused of saying (3), which, since it is true, is TO's loss. === Subject: Re: infinity > stephen@nomail.com said: > Do you think I was trying to prove that any infinite set must > have elements with an infinite number of predecessors? That > should be obvious. Is it not? It is not obvious, because it is not true. The definition of > 'infinite' does not mention predecessors. If you have some > definition of 'infinite' in your head that is based on > predecessors, please share it. > An infinite set is one with an infinite number of elements. How does one know whether a set contains an infintie number of elements? Until that is expalined satisfactorily, TO still does not have any satisfactory definitnion of a set being infinite. The Dedekind definition, on the other hand, gives an immediate and satisfactory criterion. > If the > set is ordered, every element is a successor or predecessor (not > immediate) of every other element in the set. How about sets of rationals? Not all order types are the same, and not all statements TO makes about sequentially ordered sets apply to arbitrary ordered sets. As stated TO's statement appaers to apply to any ordered set, not merely sequentially ordered sets. NOTE: a set is sequentially ordered if for every x, whenever there is a y with y > x, there is a smallest y with y >x, and similarly whenever there is a y with y < x, there is a largest y with y < x. Equivalently, between any two members of such a set there are at most finitely many other members. > Therefore, each element > either has an infinite number of predecessors, or an infinite number > of successors, or both, since all other elements are either > predecessor or successor to that element, and one cannot divide an > infinite set into two finite sets, by omitting a single element. But why cannot one split an infinite ordered set into a finite set and an infinite set by cutting it at some element? Does TO claim that the union of a finite and an infinite must be finite? You seem to have this idea that a set is infinite if and only if > there exists an element that has an infinite number of > predecessors. If you could define this notion in a non-circular > fashion, that is, define 'infinite number of predecessors' without > any reference to 'infinite set', then perhaps it would be possible > to translate your idea into standard terminology. > I defined a finite number (0 An infinite number is one > whose absolute value is greater than any finite number. Then TO cannot yet produce any evidence that any infinite number exists. > An infinite > set is one with an infinite number of elements. It doesn't get any > more basic than that. Dedekind is more basic than that. And if TO cannot define infinite set based only on set properties, he has not defined it at all, since not all sets are sets of numbers. === Subject: Re: infinity > I am trying to word things carefully, because every statement I make > gets technical objections. Everything in honest mathematics is technical, both proofs and objectins to proposed proofs. > I am trying to stick to exactly what the proof says, since it > seems have been deemed acceptable, but still not proving the case for > the entire set, based on more technical objections which really don't > make sense. Half of what is sense in TOmatics is nonsense elsewhere, since elsewhere a statement cannot be both true AND false at the same time as it can in TOmatics. Unfortunately for his mathematical development, TO seems to prefer that half. And much of what is sense in mathematics, TO labels nonsense in TOmatics, though he seems to forget that the negation of nonsense in TOmatics is equally nonsense in TOmatics. > The best way to say it, I think, is my rephrasing above. > If things aren't phrased exactly, people make all sorts of claims > about what I am saying. That is a major part of what mathematics is all about, saying things correctly! So far, TO seems to do it rarely and mostly by accident. === Subject: Re: infinity > Daryl McCullough said: Do you think I was trying to prove that any infinite set must >have elements with an infinite number of >predecessors? That should be obvious. Is it not? No, it's not obvious. The *negation* of that statement is obvious > to most people. -- > Daryl McCullough > Ithaca, NY > Only because of what they have been told, repeatedly, which is wrong. In TOmatics, what is wrong is also right, and in mathematics infinite sequences are infinite even though they have a first element. === Subject: Re: infinity > Jesse F. Hughes said: > But every element of Z has infinitely many successors and > infinitely many predecessors, so why must Z be finite? > That is simply not true if you restrict your integers to finite > values. Standard theory is wrong about that. Only in TOmatics is standard theory known to be wrong about anything, but in TOmatics, everything false is also true so that every part of standard mathematic is also true in TOmatics. Surely you can't really claim that the size of Z is twice the size > of N, since there is no such thing as the size of N. I don't see > how you can take multiples of something that doesn't exist. > You can note that you have a mirror image of N, in addition to the > original N, which makes two times whatever N is. Despite the fact > that we can't pinpoint N, we can say Z has twice the range of N. But since the range of N is non-existent, by TO's definition of range, Z's is too. How does TO multiply non-existence by two and get anything but still non-existent? > The set N has no distinct size. We know it is a finite natural, > but it would be the largest finite natural, which doesn't exist, > despite its finiteness. That's remarkable reasoning, that is. N has a size, but it's not > distinct, because if it was, it would be something that doesn't > exist, but it does, so it's not distinct. > Is there a valid objection in there, because I don't see any. TO's blindness to obvious truth in order to see the self-contradictory as truth is well documented. This is just one more instance of that well known pattern. === Subject: Re: infinity > Daryl McCullough said: It's very simple. None of the elements in the set has an infinite >number of predecessors, therefore there are not an infinite number >of elements in the set, or there would be elements with an >infinite number of predecessors. The question is *why* you think that Every element has a finite number of predecessors implies There are only finitely many elements > Because if EVERY element has a finite number of predecessors, then NO > element has an infinite number of predecessors, Okay so far. > which means there IS > no infinite sequence of numbers within the set. WRONG! It only means that there is no infinite descending sequence, but does not prohibit infinite ascending sequences. If there could not be infinite ascending sequences, TO would be requiring divergance towards negative infinity to be possible but divergence towards positive infinity to be impossible. But just multiply by -1 and that would reverse. > If there WERE an > infinite sequence of numbers, then there WOULD be elements that come > after some infinite number of other elements That is only true in TOmatics and not in stndard mathematics. In standard mathematics, an infinite sequence of all finite values can diverge upwards. > and if there are not, > then the set cannot be infinite. i am not sure why I need to explain > this. It seems obvious to me. You need to explain why what is totally false in standard mathematics must be true in TOmatics. *Why* do you think There are infinitely many elements implies There exists an element with infinitely many predecessors. > Because if the set is ordered, such that each element has a unique > index which is the successor of another index in the set, then some > elements will have infinite indexes, or positions, in the set, and > will therefore have an infinite number of elements which come before > it in the ordering. Then TO is claiming that there are no such things as infinite sequences of all finite terms that diverge upward > If no element has an infinite number of > predecessors, then there is no infinite sequence of elements in the > set. Except that the sequence of naturals by successor is an infinite sequence that diverges upward. > Notice, the answers to both questions are essentially the same. That > is because both questions are the same, being contrapositives of each > other (see, Virgil, I know what contrapositives are). The first is: > Not(exists(x: infinite(pred(x))))->not(infinite(set S)) > as opposed to: > infinite(set S)->exists(x: infinite(pred(x))) > Same question, same answer. Both equivalently false, even when constrained to the st of naturals, unless TO can show that the infinite sequence f(n) = n does not diverge upwards. Note that for an arbitrary infinite sequence, it must do one any only one of the following (1) converge to a finite value, (2) diverge upwards, (3) diverge downwards, (4) oscillate endlessly, And for a non-repeating sequence of naturals, (1), (3) and (4) are impossible. Formally, if S is some set, and S_x = the set of all y in S such > that y < x, then you are saying forall x, S_x is finite > implies > S is finite Why do you believe that implication? Is it an *assumption* on your > part, or do you believe you can prove it from more basic > assumptions? > An infintie ordered set is a sequence of an infinite number of terms, > in which sequence each element must either have an infinite number of > successors or an infinite number of predecessors, or both. However, > in the set of finite naturals, no element has either an infinite > number of predecessors or an infinite number of successors. If 1 does not have an infinite number of successors then the sequence f(n) = n, being increasing and bounded, must have a finite limit (this is a theorem about series with values in the space of real numbers), and therefore a maximal or largest value. So that TO must be caliming that what is true of those real numbers which coincide with naturals numbers is not true for ordinary natural numbers. No matter how TO wiggles, he cannot make the false into thuth, at least outside of TOmatics. === Subject: Re: infinity > I was saying, not that there is no difference, but if you say the set is > bounded, and some value in it is finite, then all values in the set are > finite. > Boundedness implies finiteness And unboundedness of a set, relative to some metric, implies infiniteness of the number of members in the set, though not in the size of any member. === Subject: Re: Why sci.math? >> And, CDC was right. There was not much support needed for the users of >> those computers with respect to the command language. >> Your mention of CDC reminded me of something. Does anyone else remember >> who the BUNCH were? (No fair looking on Google for the answer! :-) > No, but I'm going to try to guess: > Burroughs > Univac > NCR > CDC > Honeywell > Did I get any right? You got every single one right. They were IBM's major competitors, and it once was common to hear people talking about what IBM and the BUNCH were doing. (Sometimes you'll see it explained with Unisys rather than Univac, but your version is correct. The Univac system was made by Sperry-Rand, which didn't become Unisys until after it merged with Burroughs in 1986, well after the BUNCH acronym was coined.) -- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) + 1 = 0 -- Euler | -- John Myers Myers, Silverlock === Subject: linear algebra problem Problem: let A,B be symmetric positive semidefinite nxn matrices. Then tr(AB) >= 0. Any ideas? nojb. === Subject: Re: linear algebra problem > Problem: let A,B be symmetric positive semidefinite nxn matrices. Then > tr(AB) >= 0. > Any ideas? > nojb. B has a positive semidefinite square root C. Then ACC has the same trace as CAC, right? === Subject: Re: chain rule 09/15/2005 >1) The dimension of the matrix-product doesn't fit: Df(y) is a 2x1 >vektor What gives you that idea? Df is a 2x2 matrix, in this case equal to B. >2)Is the D of Df(y) a diffential against y or x. What are you trying to ask? If f:M->N, then Df(x) maps T(M)_x->T(N)_f(x). >3)How do I calculate it if I'm interested only of frac{partial >f}{partial y_1}? That depends on what f is; both g and the chain rule would be irrelevant. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: Re: chain rule | Hi there, | | I've got a little problem with the chain rule and I hope someone can | help me :-) | | Let x=(x_1,x_2), y=(y_1,y_2) and y = Bx, where B is a 2x2 matrix. B = [a b] [c d] y_1 = ax_1+bx_2 y_2 = cx_1+dx_2 | The chain rule is: | D(f(g(x))) = Df(g(x)) . Dg(x), | Thus for our example g(x)= Bx we'll get | D(f(g(x))) = Df(y) . Bx You've defined g(x) = Bx. What is f(x)? Androcles | | My questions are: | 1) The dimension of the matrix-product doesn't fit: Df(y) is a 2x1 | vektor and Bx too! | 2)Is the D of Df(y) a diffential against y or x. If it is against x, | how can I transform it to be against y? | 3)How do I calculate it if I'm interested only of frac{partial | f}{partial y_1}? | thus to the partial differtial of f against y1 | | Thanx in advance | | Richard | === Subject: Re: chain rule f: R^2 -> R is can be any derivable Function. I'm interested in the chain rule with a linear function like g(x) = Bx. === Subject: Re: chain rule | f: R^2 -> R is can be any derivable Function. I'm interested in the | chain rule with a linear function like g(x) = Bx. Ah, now I see what your difficulty is :-) g: R^2 -> R^2 because x_1 -> y_1 x_2 -> y_2 and the domain of f is then R^2 Make f(x) = Ax, A = [e f] [g h] You already know the chain rule, I presume you know matrix multiplication, f(g(x)) = ABx should be no problem. :-) Obviously f(x) = A = [1 0] [0 1] has to work. If you get into trouble, use integers and see if it makes sense. Androcles === Subject: Re: What mathematical question would you ask Judge Roberts? | Since mathematical questions are politically neutral, is there some | question that could be asked to determine if he is intelligent to be a | good justice? | | -- | Ron | Never heard of Judge Roberts, but I think you should be aware that in American law, proof beyond a reasonable doubt is required in criminal cases and proof by a preponderance of the evidence in civil cases. In mathematics, absolute incontrovertible proof is required. The small boy coming from the kitchen with chocolate on his face is strong evidence that he helped himself to the cake and is sufficient for a conviction and sentence of being sent to bed early if it's a civil case, but if he pleads that his sister put it there and she denies it then it becomes a criminal case, one of them is lying, and a mathematical proof is that YOU actually saw the event of how the chocolate got on his face. Missing cake is not enough to convict. I would ask the judge which proof he'd use to convict the small boy. Androcles. === Subject: Re: What mathematical question would you ask Judge Roberts? produce contradictions, it is necessary that the behavior > of justices not follow the rules of logical consistency. A friend who is a lawyer and also majored in computer science told me that legal reasoning was different from logical reasoning. -- Ron === Subject: Re: What mathematical question would you ask Judge Roberts? produce contradictions, it is necessary that the behavior > of justices not follow the rules of logical consistency. > A friend who is a lawyer and also majored in computer science told me > that legal reasoning was different from logical reasoning. Like I tell my classes: A mathematician uses definitions so that he knows what he's talking about. A laywer uses definitions so that he can look for loopholes. --- Christopher Heckman === Subject: Re: What mathematical question would you ask Judge Roberts? On 14 Sep 2005 22:54:46 -0700, mensanator@aol.compost > Republicans don't want justices that are intelligent, > so politically neutral questions will be evaded just like > the other ones. > So the answer is no, there is no such question. > But, being a Bush nominee, the subject of intelligence > is probably moot. Are you saying this administration's nominees do badly on a culturally-biased, meaningless test? -- http://hertzlinger.blogspot.com === Subject: Re: What mathematical question would you ask Judge Roberts? Republicans don't want justices that are intelligent, > so politically neutral questions will be evaded just like > the other ones. > So the answer is no, there is no such question. > But, being a Bush nominee, the subject of intelligence > is probably moot. > Are you saying this administration's nominees do badly on a > culturally-biased, meaningless test? What are you going to say to your kids when they ask you for help with their Intelligent Design homework? > -- > http://hertzlinger.blogspot.com === Subject: Re: What mathematical question would you ask Judge Roberts? On 14 Sep 2005 21:26:10 -0700, Ron Peterson question that could be asked to determine if he is intelligent to be > a good justice? Is it Constitutional to pass laws against non-standard analysis? Would it be a violation of Federalism to stop state governments from doing so? Did the Merck jury base their judgment on the mathematics of higher infinities? Does the administration's attitude toward budget deficits have anything to with Banach--Tarski paradox? Is it necessary for judicial decisions to be well-founded? Is strict constructionism compatible with the Axiom of Choice? -- http://hertzlinger.blogspot.com === Subject: Re: What mathematical question would you ask Judge Roberts? >Considering that one must believe that the various laws >produce contradictions, it is necessary that the behavior >of justices not follow the rules of logical consistency. > Of course the legal system is inconsistent. That is why > lawyers can argue both sides of a case (even when the facts > are not in dispute). The judge's job is to decide which > of the contradictory arguments to accept. Isn't there a story about one of the famous mathematicians of the 40's or 50's? I forget who ... but it was a foreign mathematician who was about to undergo his test for U.S. citizenship. He was eager to explain to the examiners that he had discovered a logical contradiction in the Constitution. Fortunately his friends told him not to do that! === Subject: Re: What mathematical question would you ask Judge Roberts? >>Considering that one must believe that the various laws >>produce contradictions, it is necessary that the behavior >>of justices not follow the rules of logical consistency. >> Of course the legal system is inconsistent. That is why >> lawyers can argue both sides of a case (even when the facts >> are not in dispute). The judge's job is to decide which >> of the contradictory arguments to accept. >Isn't there a story about one of the famous mathematicians of the 40's >or 50's? I forget who ... but it was a foreign mathematician who was >about to undergo his test for U.S. citizenship. He was eager to explain >to the examiners that he had discovered a logical contradiction in the >Constitution. Fortunately his friends told him not to do that! It was Godel, and he discovered a way that the US could legally become a dictatorship. See e.g. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: What mathematical question would you ask Judge Roberts? Considering that one must believe that the various laws >>produce contradictions, it is necessary that the behavior >>of justices not follow the rules of logical consistency. >> Of course the legal system is inconsistent. That is why >> lawyers can argue both sides of a case (even when the facts >> are not in dispute). The judge's job is to decide which >> of the contradictory arguments to accept. >Isn't there a story about one of the famous mathematicians of the 40's >or 50's? I forget who ... but it was a foreign mathematician who was >about to undergo his test for U.S. citizenship. He was eager to explain >to the examiners that he had discovered a logical contradiction in the >Constitution. Fortunately his friends told him not to do that! > It was Godel, and he discovered a way that the US could legally become > a dictatorship. So how can it happen? This detail is missing from the story every time I've heard it, which makes me wonder whether it's true. --- Christopher Heckman === Subject: Re: What mathematical question would you ask Judge Roberts? Isn't there a story about one of the famous mathematicians of the 40's >or 50's? I forget who ... but it was a foreign mathematician who was >about to undergo his test for U.S. citizenship. He was eager to explain >to the examiners that he had discovered a logical contradiction in the >Constitution. Fortunately his friends told him not to do that! > It was Godel, and he discovered a way that the US could legally become > a dictatorship. > So how can it happen? This detail is missing from the story every time > I've heard it, which makes me wonder whether it's true. Obviously it's true, otherwise his reasoning would have been discussed and Godel's Thesis Disproven Why is it never disclosed his reasoning? Because it's classified! The they don't want us to know what they're doing to US and to us. To find it in the constitution, I'd considering looking at clauses regarding national emergency, marshall law, war powers. Dang where's my copy of the US constitution. === Subject: Different results when doing gaussian elimination? Now and the I get different results when I do the first part of the gaussian elimination compared to maples results. On this page is an example: http://photos1.blogger.com/blogger/3626/1346/1600/gauss.jpg In the maple example, if row 3 is multiplied by 2 its the same as row 3 that I did by hand. Then I guess it only row 2 that differs. Is that allowed? My teacher told me that my by hand example was corret. Hope someone can help