mm-2459 === Subject: Re: hard task Why 4^5 is a lower bound? You don't prove that we can create 6-tuples of two participants differed in two positions, when the first 5-tuples of two participants differed in exactly one position. === Subject: Re: Continuous Injections You are surely right: my example is unnecessarily complex. However your second counterexamaple f(z,y)=(e^z,y) does not satisfy the hypotheses of the problem (soory, maybe I wasn't so clear): f must have a continuous injective extension to Cl(A) (otherwise the problem is trivial!). What is WLOG f? For the same reason, this counterexample is not acceptable (I think that it will be particularly difficult to find a counterexample in the C^0 case, if it exists). Maury === Subject: Re: dense set having a nested sequence of open intervals around the point in question, with lengths converging to 0. Each interval will contain a point of the dense set. The sequence of these points will converge to the point of question. There may be a distinction if you make the axiom of choice an issue: given the open onterval, how do I pick a point from D in it, and how do I perform a countable number of choices in one stroke? For D being the set of rational numbers, there is a way out: the rationals can be arranged in an explicit sequence (repetitions allowed), and then we pick the first rational that falls into the given interval. But if D is not countable (such as the non-computable reals - and there are uncountably many of those, and they are dense in the reals), picking a convergent sequence is a pure matter of faith. === Subject: integral transformation and parametric curves I'm looking at the following example B=[5,2;0 3] (rowwise defined matrix). This transformation convert a unit square in a parallelogramm. I tried to calculate the normal derivative of f on the edges. I began with the left one, let's name him E1, where E1=B(e1). e1 ist left vertical edge of the unit square. I got: n = (n1,n2) ist the normal vector of E1 and n^T is the transpose of n. int_E1 (partial_n f(X))^2 dX = |det B| int_e1 ((f(g(x)) . B^(-1) .n^T))^2 dx can't be right because |det B| describes a transformation of an area and hier we got a transformation of a length. I think it should be something like the lenght of E1 but how to proof it? my idea was to use parametric curves but I did get it :-(( Please give me a hint Richard === Subject: Analogue of Noether's Theorem for finite groups? Noether's Theorem is considered by many physicists the most important theorem pertaining to physics. It basically says that for any symmetry, there is a corresponding quantity that is conserved. For instance, because the laws of physics are the same at every moment in time, we have the law of conservation of energy. And because the laws of physics are the same for each position in space, we have the law of conservation of momentum. Is there an analogous theory for finite group actions or for at least some types of finite group actions? Craig === Subject: Re: Analogue of Noether's Theorem for finite groups? Well... It's a little more complicated than that. Basically, you have to be able to describe a conserved current. So, for example, you can relate time translation invariance of the Hamiltonian to energy currents. Basically you can show that the energy flowing in/out of a volume is related to the amount of energy in the volume by an equation that gives a conservation law. Basically the time deriv of the total energy in the volume is related to the integral of the divergence of the energy current over the surface of the volume. And hey, presto, you've got Stokes theorm in there. You can relate volume and surface integrals in certain cases. Similarly with space translation symmetry, you can show that momentum, the thing canonically conjugate to position through the Lagrangian, is related to a conservation equation of momentum in a volume related to momentum leaving/entering that volume. In very similar fashion, you can get a conservation equation for electric charge. In that case, the name conserved current is probably a little more familiar. The time deriv of charge inside a volume is related to the rate at which charge is entering/leaving the volume. And that is related to the divergence of a current integrated over the surface. In this case, the symmetry is gauge symmetry. But the symmetry alone isn't enough. You also need a conserved current and its time behaviour, i.e., a Lagrangian or Hamiltonian. Not directly at any rate. There might be something in cases where there is a continuous limit to be taken. For example, in lattice gauge theory, you still need to conserve your conserved quantities. This is true even when your lattice is at a fairly small number of points. It would not do for a lattice gauge calculation to spontaneously generate new electric charge. However, you probably achieve this by other means rather than by using an analogue of N's theorm. That is, you probably start with the continuous form of N's theorm and discretize it, rather than starting with a finite group and deriving an analogue of N's theorm with the right limit. Certainly one needs to worry about conservation laws in numerical situations. When performing numerical fluid dynamics calculations, for example, one needs to take care that the various conserved quantities are conserved at least up to the accuracy of the solution method. Again, though, it is likely you start with the conservation law in continuous form and derive a discrete version with the correct continuous limit, rather than starting with a discrete conservation law. Socks === Subject: measurable functions Hi Q: If f is a function on A U B (U is union), show that f is measurable if and only if f is measurable on A and on B. A and B both measurable sets. I think I got one way: if f is measurable on D and on E then by definition the sets {x: f(x) < a} for x in A and a real is measurable and sets {x: f(x) < a} for x in B and a real is measurable therefore their union is {x: f(x) < a} for x in (A U B) and a real. As a union of measurable sets is measurable then f is measurable on (A U B). Is this right? What about the other way? Mark === Subject: Re: measurable functions Do you know that the space of measurable functions is a ring? -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: measurable functions Yes I know it is a ring. But the question here isnt about f+g or f*g, its a question about the domain on which f is measurable. === Subject: Re: measurable functions How about letting g be an appropriate characteristic function? === Subject: Re: measurable functions What about: Let fA be f restricted to A, and fB be f restricted B, and & be intersection of sets. Then ((fA)^(-1))(a,inf] = (f^(-1))(a,inf]& A then fA is measurable? Would this be right? And is the first direction that I showed right? === Subject: Re: Standard Deviation of PISA Well done. Practically a master's thesis. I predict horizon-to-horizon herds of wombats. -- cary === Subject: Re: Function discretization [CORRECT] ??? x2-x1= (y2-y1)/(x2-x1) this is not he minimum mean squrae error of two values y1 y2 that is simpy (y1+y2)/2 too much conditions: xleft =x1 xright=x11 mid points (x_{i+1}+x{i})/2 must be tabulated points (y_{i+1}+y_i)/2*(x_{i+1}-x_i) independent of i i=1,...10, x_{i+1}-x_i = (y_{i+1}-y_{i})/(x_{i+1}-x_i) i=1,..,10 I guess you mean something different: the so called histospline (approximation of a funtion by a step function such that rectangle areas of the pieces are equal) hth peter === Subject: Re: Function discretization Peter Spellucci ha scritto: I had enough time to get more focused on the problem i.e.: keep the sum of the area of the rectangles equal to the integral of the function. Is there any tentative solution to this problem? === Subject: Re: Rational vs irrational Actually, it means the reference used in Mathworld was from 1998. Not that the first time it was published was 1998. Often, the most convenient reference for something isn't its first publication. === Subject: Re: Rational vs irrational If it's expressible, you should be able to write down a finite, unambiguous specification for it. A simple counting argument then sufficies to show that not every real is expressible. Yes, every real is the limit of a sequence of rationals. But not every sequence of rationals is expressible. === Subject: Re: Sphere inside a pyramid Should be only when the tetrahedral space is a quadrilateral extruded prism with parallel tetrahedron edges? === Subject: Re: Sphere inside a pyramid Should be only when the tetrahedral space is a quadrilateral extruded prism with parallel tetrahedron edges? === Subject: Re: Sphere inside a pyramid What was missing!? === Subject: Invariant polynomials let G be a finite 4x4 matrix group. We can define an action of G over the vectorial space of the polynomials in coord. (x,y,z,w) in the usual way: (gP)(x,y,z,w) := P(g(x,y,z,w)) with g in G and P a polynomial. I remenber that a G-invariant polynomial is a polynomial P s.t. gP=P for all g in G. How can I compute, with Maple or other sw, a basis of the vectorial space of G-invariant polynomials with degree less or equal to n? === Subject: Re: Invariant polynomials G corresponds to a set of linear maps on the finite-dimensional space V of polynomials of degree <= n. Choose a basis of V (presumably consisting of the monomials x^j y^k z^l w^m), and these maps are represented by matrices. You want a basis of the intersection of the kernels of the matrices representing g - identity for g in G (of course a set of generators would suffice, rather than the whole of G). In Maple, I would use GenerateMatrix, NullSpace and IntersectionBasis in the LinearAlgebra package. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Invariant polynomials On 2005-09-23 21:09:39 +0200, israel@math.ubc.ca (Robert Israel) said: thank you for your answer but I'm not sure my question was well understood (or, vice-versa, I'm not sure to understand what you said...). Of course I want a basis for the intersection of the kernels but they are kernels with respect to the action of G over the space of polynomials. Not the kernels in the usual linear algebra sense. Is this computed by NullSpace? Can you describe more exactly what I have to do? I have two generators Bye, Nicola === Subject: Re: Integration of x^n * Exp[-x^2] No, the _indefinite_ integral cannot be expressed in a finite number of terms involving elementary functions. (Elementary functions are sums, products, powers, trigonometric functions, exponentials, and the inverses of all these. They are the ones you have seen in high school and introductory Calculus.) Note that this is a THEOREM: it really is *impossible* to do the integral in finitely many elementary operations. It is not just that nobody has yet discovered the formula, but, rather, that no such formula is possible. A few years ago, this topic was discussed in detail in this newsgroup, and an actual proof was supplied. Of course, we can do the definite integral from 0 to infinity, or from -infinity to infinity. Ray Vickson Adjunct Professor, University of Waterloo. === Subject: Re: Where is my error? Nntp-Posting-Host: hera.cwi.nl In the future pray allow some context with your follow-up. But indeed. Given that ln(-1) = {(2n + 1).i.pi} we find that 2*ln(-1) = {(4n + 2).i.pi}. But ln(1) = {2n.i.pi}. So 2*ln(-1) != ln(1). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Where is my error? True, 2*ln(-1) and ln((-1)^2) are not the same, and so p*ln(b) = ln(b^p) fails here. However, it might be interesting to note that ln(-1) + ln(-1) and ln((-1).(-1)) _are_ the same and so ln(a) + ln(b) = ln(a.b) still holds here. Of course, this brings to light the fact that 2*ln(-1) and ln(-1) + ln(-1) are not the same. Those unfamiliar with doing arithmetic with sets of values might find that fact to be rather curious. David Cantrell === Subject: Re: determining if point resides on a line segment There is a more elegant way. In exact arithmetic [ 1 1 1 ] 2A= det [ xa xb xc ] = 0 [ ya yb yc ] A is area of triangle formed by a,b,c. In inexact arithmetic a tolerance has to be introduced. === Subject: Re: determining if point resides on a line segment I don't see how this will work. The triangle's area will be zero whenever c lies anywhere on the extended line through a and b. We want to test whether it's actually on the segment between a and b. === Subject: Re: determining if point resides on a line segment Didnt read the OQ carefully. OK, A=0 is necessary but not sufficient. Check a=c || b=c; if fail then compute dot products (b-a).(c-a) and (b-a).(c-b) & check for sign reversal. One of my students has another idea: first check A=0, then take c = alfa a + beta b (a,b,c being position 2-vectors, alfa & beta scalars), solve for alfa & beta, check for positivity. === Subject: Re: determining if point resides on a line segment Didnt read the OQ carefully. OK, A=0 is necessary but not sufficient. Check a=c || b=c; if fail then compute dot products (b-a).(c-a) and (b-a).(c-b) & check for sign reversal. === Subject: Re: anything to the power of 0 = 1, why???? A^0 = 1 except for A = 0. We know A^1, A^2, A^3, ...; We also know A^(-1), A^(-2)...; But it is not easy to imagine what A^0 is. Let us put down what we know A^3 = A * A * A A^2 = A^3 / A = A * A A^1 = A^2 / A = A A^0 = A^1 / A = 1 (Please note : We define A^0 = 1 so that it would be consistent with other results.) A^(-1) = A^0 / A = 1 / A A^(-2) = A^(-1) / A = 1 / A^2 === Subject: Re: anything to the power of 0 = 1, why???? tried to post and got an error. Since the error may have left her thinking that her post didn't go through, she may have backed up and attempted to repost. It happens. === Subject: Re: anything to the power of 0 = 1, why???? Fernando. === Subject: Re: anything to the power of 0 = 1, why???? Tom, don't worry, this is the (n^n)! th time that nonton has posted this topic, so I'm afraid that his(her) only intention is to examine to all members of sci.math. For that reason, nonton never answers, he(she) is is The Teacher. Fernando. === Subject: Re: binomial expansion ok, that is good to hear. I was mainly worried about switching x with (a/b)^2, where x is linear and (a/b)^2 is not. But I guess it is ok to do that. === Subject: Re: binomial expansion Yes, x is linear wrt x, but then so is (a/b)^2 linear wrt (a/b)^2. This is not meant to be a flippant answer; I'm hoping that you'll gain some insight about what we're really doing here. The fact that we treat some variable x as independent does not imply we *couldn't* define some other equation specifying how x depends on other variables (even nonlinearly). It only means that for the purposes of the manipulations we are doing, those other dependencies are irrelevant. === Subject: Re: differential equation Maple can't solve it, and finds no symmetries. I doubt that there's a closed-form solution other than the constants f = n pi. Of course you can solve it numerically, or using series. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: dimension of intersection of subspaces. i get your point, but how do i exactly prove that? can you give more hints, i could solve the first part of the problem. thanx wood === Subject: elements of odd order Let G be a group and let a,b be elements of G. Suppose that a has odd order and that aba^(-1)=b^(-1), show that b^2=e. I'm stuck, any ideas? === Subject: Re: elements of odd order Note that a^(2)ba^(-2)=ab^(-1)a^(-1)=(aba^(-1))^(-1) =(b^(-1))^(-1) =b. Also that a^(3)ba^(-3)=aba^(-1)=b^(-1). Establish that for n even a^nba^(-n)=b and for n odd a^nba^(-n)=b^(-1). Muhammad === Subject: Re: Revisiting my (alleged) proof of the Goldbach Conjecture Well, that is why I was asking you whether there is a Goldbach look-alike, whether there is a Perfect Number look-alike, and whether (a future question) there is a Riemann Hypothesis look-alike. I know there are not enough primes but since we do not have operations between say 3-adics and 10-adics whether in the defining of operations, whether there is a Goldbach look-alike or a Perfect Number look-alike or a Reimann Hypothesis look alike. I disagree with you Dik, by ignoring does make it ill-defined. Suppose the 4 Color Mapping were true, then the Jordan Curve theorem is false, since 4CM implies 4 colors are necessary whereas JCT implies 2 colors are necessary. Chessboards and checkerboards would have to be 4 colors. Black and white photographs would be unintelligible and a picture to be intelligible would require a minimum of 4 colors. If the 4CM were true then language would not exist as we know it today because all language can be based on 0 and 1 coding. Computers would not exist today since electronics is based on whether the gate is open or closed which is the Jordan Curve theorem. If 4CM were true then Computers would have to be based on 4 gates, not 2 gates. It is said in Mathematics that a thing of falsehood cannot exist because it contradicts the whole of mathematics. The 4 Color Mapping is false because it says that 4 colors are necessary when the Jordan Curve theorem says that 2 colors are necessary. Both of them cannot be true. If I understand this Sigma definition, then 1 is perfect because 1 + 1 = 2. However, I do not understand how this Sigma treats those numbers such as 9 and 16 etc etc. Whether 9 is 1 + 3 + 3 + 9, or whether 9 is something else. So, Dik, can you clarify how Sigma treats 9,16, 25, etc etc Okay, call me hardheaded, but you also display a bit of hardheadedness when it comes to sticking with all the old literature of mathematics and when someone seems to be attacking the old ideas, you seem to be overzealously defending the old ideas and old definitions. So you should keep in mind that you are also hardheaded. Another example is the Adics, whenever I ask about the Adics, your answers are always predictable in that you reveal what the questioner knows in the first place, then you pretend as to not know what the word Adics means. And you finally exit the thread by saying the Adics are a loose collection with undefined operations. When it is exactly the clarification of whether those operations can have a Goldbach look-alike or have a Perfect Number look-alike or have a Riemann Hypothesis look-alike. 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 Why is the date time group all messed up in two posts this morning. The above post to Dik shows a date time group of 23 Sep 2005 12:05:54 -0700 and an hour later I made another post to this thread to Chris and that hour later post has a date time group of 23 Sep 2005 12:02:53 -0700. So does the ISP and Google or someone have a clock that is not in sync? Or is a hacker responsible? Or what is the cause of this error? Such an error is hard on archival material where the order is very important and the correct date and time is very important. Such an error could really ruin a archive 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 Yes it is. A graph G is planar if there is an injective embedding E (best expressed as a set of embeddings) into the plane (R^2) with the property: For every edge e=uv in G, the embedding of that edge is a piecewise linear function E_e from [0,1] to the plane R^2, and either E(u) = E_e(0) and E(v) = E_e(1) or vice versa. This is the formalization of G can be drawn in the plane without edges crossing. The characterization of planar graphs by Kuratowski means that we can also look at planar graphs without considering an actual embedding. A.P. Well I am not familar with this claimed equivalent statement. But it sounds far removed from that of 4CM which simply says 4 colors are necessary and sufficient to color every map. So, Chris, can you please translate this Vertice statement into some common everyday world experience. The fear I have with these so called equivalent statements since 4 CM is a fake, is that the equivalent statements have to be so far removed from any reality because the underlying statement is a fake. I posted to Dik earlier and I should not repeat what I said there. But a simple test of 4CM is that of electronics and computers. They are based on the Jordan Curve theorem that 2 colors suffice. Electronics needs 2 and only 2 gates, open or closed, in fact the 2-adics can serve as the foundation of Electronics and Computers. But if the 4 Color Mapping is true then 4 colors are necessary and 4 gates are necessary. In fact if 4CM were true then black and white photographs would be unintelligible and lack information, but we all know that black and white contains all the information that color photography has. The first one. You look at the SET of all positive divisors of n (except n), and add up the elements of that set. In sets, the number of times that an element appears is irrelevant; each element is added once. So the sum of the proper divisors of 9 is 1+3 = 4. And this is how the rest of the world (everyone other than you) defines whether a number is perfect or not. If you're going to repeat elements of that set, you may as well write 9 = 1 + 3 + 3 + 1 + 1 and conclude that 9 is perfect after all. A.P: Dik gives a Sigma definition where the sum is 2n. For the number 9, I believe the correct definition would be this 1 + 3 + 3 + 9. Chris and Dik seem to believe for 9 the correct form is 1 + 3 + 9. The reason I disagree is because the divisors of a number n are governed by the square root of that number so that all the divisors have a value of the square root or less and there are no other values to hunt for and those values are paired. So for 18 the square root is 4.2 and so all I have to worry about is 1,2,3,4 and if one of them has a divisor it is paired so that 1 is paired to 18 and 2 is paired to 9 and 3 is paired to 6. So here is the ambiguity for Chris and Dik, are we to treat the definition of Perfect for 9 as that of 1 + 3 + 9 or are we to treat the definition as 1 + 3 + 3 + 9. Chris seems to think I randomly is adding numbers but I am not randomly doing that. 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: question about subspaces hi i have a question here. prove that dim(W1+W2)<=m+n === Subject: Re: question about subspaces This looks like a homework problem that you are asking others to do for you. Show us what you have done so far; then maybe we would be willing to help, if there is some point you are having difficulty with. Ray Vickson === Subject: Re: question about subspaces So far he's extracted an almost complete solution from other people in a different thread, and is angling for more help. === Subject: Re: question about subspaces ya, its homework problem, consider the set S, of basis vectors of W1 and W2. then every vector in W1+W2 should be able to express as a linear combination of S. does this mean that dim(W1+W2) should be equal to cardinality of S ? === Subject: Re: a simple question on algebra (left/right identities and left/right inverses). identity literally 'stares at you'. I assume I can easily remap G^2 (for set with 2 elements) to G^1 (for set with 4 elements). === Subject: Re: a simple question on algebra (left/right identities and left/right inverses). Yes, that was my question, and sorry I didn't make it really clear. I have to learn to be less verbose, I guess. The only thing to add is that since it does follow that for all a in S, there is no a' such that a'a = e, the counterexample would involve the structure with left identity and no inverses whatsoever (neither left nor right). === Subject: Re: a simple question on algebra (left/right identities and left/right inverses). There is a still weaker version of identity... For every a, there exists u such that au=ua=a. And for inverse, continue: and there exists b such that ab=ba=u. Then try to prove that u is the same for all a... -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: A higher math education As I am now jobless and have an interest in math, I am seriously considering pursuing a Master's or Doctorate in Math. I would most likely want to become a professor. is Computers, my math goes up to Calculus I. So the Math Department Director says, I have to take 5 Math (Calc II & III, Linear Algebra (AGAIN) and Calculus Of Variations I & II) before getting started in the Master's program. As it turns out, due to requisites, I have to CalcVar II... I can take LA anytime after Calc II). This is too slow! I suppose I can take a job and take the classes concurrently anyway but really, in the modern era I say to myself, it doesn't have to take that long. So I wonder about programs out there where a person like me (no job, so all time would be dedicated to studies) can enroll and graduate fast, by offering the degree in shorter terms (i.e., more hours during less days). I'm a tech-head and I have no issue doing this stuff for many hours in the same day, after all, what is expected of a professional in the workplace anyway. I know both English and Spanish very fluently. Heck, I would even consider learning a new language! So, any suggestions? I would consider any country in the world, BTW. Oh and wish me luck, thank you. :) === Subject: Re: A higher math education I don't know what the modernity of the era has got to do with the speed at which one learns mathematics. (Did the ancient Egyptians, admiring their pyramids, think of themselves as living in the modern era? Probably.) In my country, you wouldn't get on a Masters course without a first or 2:1 in the subject. (Not some other subject, a degree in computing, no matter how good, wouldn't get you onto a masters course in mathematics. (Generally, but see the quotation below.)) While you're spending your four terms doing calculus, linear algebra and calculus of variations why not do some self study in parallel? In that way the four terms might not seem so slow. What you'd do would depend on what area you wanted to specialize in. If analysis, then topology, measure theory, functional analysis, for example; if algebra, then Galois theory, rings and modules, category theory, for example. French, if any. Do Part III of the Mathematical Tripos at Cambridge. One year. http://www.maths.cam.ac.uk/CASM/: The Certificate is not an easy course. Non-Cambridge graduates are normally required to have a first class honours degree in mathematics, physics, or engineering, or an equivalent qualification. Candidates from within Cambridge are normally required to have obtained first class honours, or very good second class honours. I think you need good sense more than good luck. -- I don't know who you are Sir, or where you come from, but you've done me a power of good. === Subject: Re: A higher math education Once you're in the grad program, you should qualify for an assistantship. A single guy can live decently on the typical assistantship (around $12K per year plus tuition waver.) Check with your department. Bart === Subject: Re: A higher math education You will never get an academic position with just a Master's degree. What I am about to say is not a reflection on you. This *really* surprises me. I am astonished that any graduate program would accept such a limited preparatory backgound. (1) The calculus course are probably not proof-oriented. Correct? How much theorem proving have you done? (2) At a *minimum*, the following is missing (for any decent graduate program!) A year of Abstract Algebra A semester of general topology A semester of real analysis (not just calculus) A semester of complex analysis (3) I know of NO graduate program that requires Calculus of Variations. If your background consists only of the courses you mentioned it will take a lot longer than you think. With the stated background, a PhD will take you a minimum of 5 years full time. I do wish you luck. Go for it! === Subject: Re: A higher math education In my country he would struggle with a doctorate. He'd need a good research record as well. A doctorate might get him a one, two or three-year research fellowship during which he may be able to earn a few pennies teaching a course or two. -- I don't know who you are Sir, or where you come from, but you've done me a power of good. === Subject: Re: A higher math education Math is the slowest Ph.D. you can get. People of average ability (average for a math person) typically take 5 years or more in grad school. One guy I went to grad school with had his thesis topic published by someone else twice, right before he was finished. Each time he had to start all over again. One suggestion would be for you to complete a Master's degree, which you can do in a couple of years. Then you can get a job teaching math in a community college. === Subject: Re: A higher math education , That is seriously misleading. === Subject: Re: A higher math education I don't agree with this. Perhaps it is slower than the sciences, but in humanities doctorates tend to take longer. At U.C. Berkeley, the average duration of a Ph.D. in Egyptology was eleven years! -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor in Central New Jersey and Manhattan === Subject: Re: infinity Virgil said: AHHH ha ha ha ha ah ah hahahaha! Good one! cardinality..... -- Smiles, Tony === Subject: Re: infinity David R Tribble said: Yes, 0:500...000. It works just like the finite unit. half of 1.0 is 0.5 (assuming decimal) I imagine you wanted to add 8080...8080 to get 999...999? Otherwise you get 1:0101...010100 Not necessarily. In this case it would appear so. In the case of 123123...123123, you really couldn't say. Not necessarily, but it would be equal to 1:011011...011010, which tells us nothing about whether the number of digits is even or odd. Is aleph_0 even or odd? -- Smiles, Tony === Subject: Re: infinity aleph_0 is not a natural number, and only natural numbers are even or odd, in the sense of being divisible by two or not. I do not know about the things that you mistakenly call 'natural numbers', but every natural number is even or odd. Stephen === Subject: Re: infinity Simple. Put ten of them into a jar, then remove the earliest one you put in and eat it. Repeat an infinite number of times until noon. === Subject: Re: infinity Virgil said: Except for maybe sum(). How about sum? Remember our discussions of a sum of an infinite number of 1's? Funny that should come up again from yet another source. -- Smiles, Tony === Subject: Re: infinity David R Tribble said: I don't believe so. I can't imagine any such mapping, and I would imagine that if it existed, then it could be turned into a mapping onto all the reals. That is only given that N is an infinite set. If N is infinite then, yes, you can map the bit strings denoting membership in the subsets of N to the bit strings denoting the values of reals in [0,1]. However, as you know, it is my position that N is not infinite without infinite n in N. If this is the case, and N includes infinite naturals, then these infinite naturals are also represented by infinite binary strings. In this case, we can draw a bijection between the reals in [0,1] and the elements of P(N), since those elements will have a truly infinite number of bits. Of course, if this is the case, we can also draw a bijection between the reals in [0,1] and the naturals in [1,oo] through their infinite bit strings, as well as between the infinite bit strings in N and those in P(N). If you include infinite naturals, then you can create this bijection between the set and its own power set, which clearly shows that bijection alone does not indicate equal sets. This is probably the root reason why naturals are not allowed to be infinite. -- Smiles, Tony === Subject: Re: infinity David R Tribble said: Not quite. You can define a bijection between *N and P(N), but you still can't define a bijection between *N and P(*N). You can't define a bijection between a set A and its power set P(A) for any set A, regardless of whether it's finite or infinite, and regardless of whether it contains infinite members or not. === Subject: Re: infinity If *N means the set of all strings of digits, finite or infinite, then it's fairly easy to construct the mapping onto [0,1] and thus onto R. Just reverse the digit order and interpret the string as a decimal in [0,1]. - Randy === Subject: Re: infinity I think it's the non-standard (a la Robinson) integers. But in any case, it's still fairly easy to get a mapping from *N onto [0,1]. === Subject: Re: infinity David R Tribble said: More than infinite? What do you mean? If N is infinite, 10N is infinite. You did this N times, with 10 balls each time. Where did the balls come from? I dunno, the infinite ball store? You have an infinite number of reals in [0,1]. How can we have more than an infinite number of reals? We can have ten times this infinite number, in [0,10]. -- Smiles, Tony === Subject: Re: infinity David R Tribble said: Except, of course, we can't. For every x in [0,1] there is a corresponding 10x in [0,10]. If [0,10] has ten times as many elements as [0,1], and we count every x in [0,1] and every corresponding 10x in [0,10], which points did we skip? === Subject: Re: infinity David Kastrup said: Yes, well, you have an issue here in that 0 is its own successor. Okay, it's recursively defined, and not infinite. Point taken. Nice trick. Why didn't you just say 0 is an element if x is an element x+0 is an element. I guess that wouldn;t be tricky enough. -- Smiles, Tony === Subject: Re: infinity David Kastrup said: What do you use it for? -- Smiles, Tony === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ It's sufficient to put it out in the right channels, and it will eventually end up being applied in a fertile way. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity Randy Poe said: {2,4,6,...,2N} not in {1,2,3,....,N}? In that case, there are no finite elements in the first that are not in the second, given infinite N. It has only half the elements up to N, and the other half are larger than N. That sure would be bad. Sure am glad it's not the case! If N is infinite, the range is infinite. I have not restricted myself to finite numbers. If you are restricting yourself to finite numbers, then you have a problem. If your set of naturals contains ALL finite naturals, then you now have elements which are twice as big, so the top half of those are now larger than any of your original finite naturals. That would seem to be a bad thing. -- Smiles, Tony === Subject: Re: infinity stephen@nomail.com said: But, but, brains don't have sensory neurons!!! It must be your sinuses, or neck or something. No pain no gain, but if you want to, have an aspirin. Anyway, that is about right. The size of a set that never ends is really not identifiable. Sets can be compared for size over some common range, finite or infinite. Over the infinite unending range, infinity=infinity=infinity, which is the way most common folk think of it. In order to get a finer view with any real consistency, one has to look at some range. If you say ALL the evens vs. ALL the naturals, that assumes some common overall range, and one can say there are half as many. If you want to make a bijection based on f(x)=2*x, whenever you apply such a mapping to any set, you double the value range, which standard theory ignores when declaring equal set sizes based on the bijection. -- Smiles, Tony === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ Don't think so hard, Tony, or you'll hurt your sinuses. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity Nope. When I say ALL the evens, I mean ALL the evens, and when I say ALL the naturals, I mean ALL the naturals. There is no assumption of range at all. I do not think of 'range' at all. It is obvious that when you say ALL you really do not mean ALL, but do not assume that everyone else has that same impairment. As near as I can tell, you simply cannot conceive of a set unless both ends have been identified. Unending sets are apparently beyond the scope of your imagination. Stephen === Subject: Re: infinity David R Tribble said: If E and T have the same value range, they are the same set. That should be clear. Value range is crucial to determing realtive sizes of infinite quantitative sets in Bigulosity. -- Smiles, Tony === Subject: Re: infinity David R Tribble said: So do E and T have the same value range or not? Are they the same size or not? === Subject: Re: infinity Is this true for any arbitrary pair of sets? Matt === Subject: Re: infinity stephen@nomail.com said: If value range is the maximum difference between elements, and all elements are finite, then all differences are finite, and the range is finite. Does it matter whether we can identify the largest possible difference? No, it is sufficient to know that all differences are finite in order to know the range is finite. It is necessary to know the largest and smallest to know exactly what the range is, but not to know that it is not infinite. And you have not explained how you have a set of values that increases forever until you have an infinite number of values, but they never get infinite, even though they increase forever. If I have any set of values, knowing only that they are finite and nothing else, can they have an infinite range of values? Can ANY pair be infinitely different? No, the range must be finite, even if I don't know what it is exactly. Do you disagree with that statement? I use my definition of finite by saying -- Smiles, Tony === Subject: Re: infinity No. Because the range is the greatest difference between elements *provided such a difference exists*. But I can find (say) the least natural number, and given any natural number I can find a greater number, which will have a greater distance from the least number. Therefore, there is no greatest distance between elements, and thus no range. Matt === Subject: Re: infinity Not if the maximum does not exist. Yes it does matter. You cannot claim that non existent objects have properties. And before you start talking about witchs and unicorns, I am not talking about physical existence, but mathematical existence. No, that is not sufficient. That maybe an axiom of yours, but otherwise it does not logically follow from anything. You do not know what the range is. It is some non-existent thing. Claiming that it therefore must be finite is not a proof. I and others have explained it dozens of times. You even agreed to it in another post. Here it is again: For each finite n, the list {1, 2, ... n } has a last element. If a list has a last element, it is finite. The list { {1}, {1, 2}, {1, 2, 3}, ...... does not have a last element. If a list does not have a last element, it is infinite. And again, you keep assuming that people think the set of natural numbers is created by adding elements one at a time. You say we have a set of values that increases forever until you have an infinite number of values. The set of values does not 'increase forever until'. The set of values is declared to all exist simultaneously. There is no time involved. The set never changes. Of course I disagree with that statment, assuming range is defined as the maximum difference. There is no maximum difference. A non-existent difference is not finite. It is not infinite either. It is non-existent. It has no properties. But you did not use your definitions to decide that there was a finite value range. You invoked your mysterious 'non existent elements have properties' axiom. 'finite' could be an entirely undefined term, and your 'argument' still works. The only axioms you use are: if x is finite, and y is finite, then x-y is finite if x is finite, and y is finite, then x/y is finite the maximum element of a set of finite objects is finite, even if there is no maximum element Of course you have never actually stated that last axiom, and it leads to immmediate contradictions which you always ignore. Stephen === Subject: Re: infinity albstorz@gmx.de said: Because they are scared of them. They had a big fracas in the 1800's. Many people still wanted infinity to be mysterious, so we are not allowed to talk about infinite numbers. It's a religious thing. Luckily for me, though I seek spirituality, I don't join religions. I agree with you. Numbers represent sets, ans set sizes are numbers. Infinite sets have sizes which are infinite numbers. I don't see the difference either. :D -- Smiles, Tony === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ N = 1/0 = 1/(2*0) = (1/0)/2 = N/2 You are babbling words you don't understand. What do they quantify with regard to set sizes. You have fantasized about your idea being applicable, but you have shown no connection except for wishful thinking. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ Tony, you are turning over stones in the closet, not in the wilderness. Your competence is that of a child, and you don't know the difference. Tony, bumbling headlong into walls is not exploring. The problem is not that you are venturing on new ground. The problem is that you are on old ground, ground that has been covered by many beginners. There is nothing new or original in your ramblings. And most particular, there is nothing working in it. You think the wagonloads of self-contradictions and non-sequiturs you are in are just pioneering problems. But all your stuff is just naive stuff on the level of a child, and every mathematician with just a tiny bit of education can poke holes in all of your arguments. Now again, if you want to make a self-delusional fool out of yourself, no harm done. But Martin _is_ harming his career if he does the same. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity David R Tribble said: There is no largest N, finite or infinite. N is a unit infinity, or rather, a variable which can assume infinite as well as finite values. It's an arbitrary largest value in the set. At least that is the way I use it. -- Smiles, Tony === Subject: Re: infinity What is an 'arbitrary largest value'? I have no idea what that can possibly mean. If N is not the largest value, you should just call it an 'arbitrary value'. Claiming that there is no largest, and then two sentences latter saying something is an 'arbitrary largest value' is silly. Can you define any of these terms in first order logic? Here is a definition for z is the largest value in S, using 'A' to mean 'for all'. What is the definition for z is an arbitrary largest value in S? Stephen === Subject: Re: infinity Virgil said: You mean the naturals are not closed under division, but the reals are, if you include the infinites. Otherwise you have to exclude zero as a denominator. Yes, infintiesimal is the correct term, but when I used it before, I was told there was no such thing. It's a little ahrder for you to say that about zero. Are you over your head? Here have a floatie tube. Uh, no. The deepest truths are approached by analogy. Give me back my floatie! Proof is not always the goal of explanation. -- Smiles, Tony === Subject: Re: infinity Jonathan Hoyle said: Uh, do you expect me to prove that the set of all strings, finite and infinite, is finite? It's not. I don't know what you want. No matter what finite size you have, the set of strings is always finite. That's what was proven. So, you're God? Wow! I never thought I'd meet you in a place like this. Gee, talking with God about infinity! I hope I'm not going to hell for this! I'm not, am I Jonathan, uh, I mean, God? Please? I'm sorry. Please don't send me to hell. Pleeeease! My reasoning is not circular, O Infinite One, and if you continue to say that, then that continuation consititutes the main circularity of the discussion. Watch carefully: Can you please point out the implication that points back to the beginning and causes a circle? Can you identify which of those two linear implications you disagree with? No, it doesn't bother me for my sake that you disagree with me because you see circles that aren't there. It also doesn't make me feel bad for me that cops don't understand keeping peace, doctors don't understand health, lawyers don't understand justice, and politicians don't understand the role of law. It just sucks for the world at large. -- Smiles, Tony === Subject: Re: infinity Clearly the last one. You claim that the range equals the maximum finite difference. There is no maximium finite difference. Therefore there is no range. Yes, the fact that all values are finite means that all differences are finite. But if you define range as 'maximum finite difference', then the fact that all diffences are finite does not imply a finite value range. You apparently have some other hidden axiom somewhere that says that if the maximum finite difference does not exist, then it is finite, but you have never stated this axiom. Your theory depends on a lot on non-existent objects, so you really need to formalize the properties of those non-existent objects. Here is a question for you Tony: is the following statement true? if x is a finite natural, then there exists a finite natural Stephen === Subject: Re: infinity Consider the sets: A_0 = {aleph_0} A_1 = {aleph_1} A_2 = {aleph_2} ... A_i = {aleph_i} for all natural i Each A_i is a finite set (a finite value) with exactly one member. Now we define an ordering relation between these sets: A_j < A_k iff j < k, A_j = A_k iff j = k. And we define a difference relation between these sets: A_j - A_k = j - k for all j <= k. Now consider the set: A = A_0 u A_1 u A_2 u ... A is an infinite (unbounded) set containing every A_i set, and every member of A is a finite (one-member) set. The difference between any two members of A (i.e., A_j - A_k, for all j and k) is a finite number. We can define a one-to-one mapping between A and N: f(k) = A_k for all natural k which means that there is a set A_k for every natural k, which means that A has exactly the same number of members as N. How big is A? What is its value range? === Subject: Re: infinity Daryl McCullough said: LOL! You don't have to pay attention. Yes, I need to go away and write it all down in a coherent manner. This haggling doesn't convince anyone. However, it does prepare me for the kinds of objections and arguments I am likely to face, and gives me a better focus on what the issues are with the standard theory, and how to correct them. Sure, it's a work in progress. Discussing it has taught me a lot. Has it taught me I'm wrong? Not at all. Will I go away? Not on command. -- Smiles, Tony === Subject: Re: infinity David R Tribble said: If two sets have the same number of elements, are they not the same size? === Subject: Re: infinity Virgil said: Yes, here you have defined infinite predecessors as you do infinite successors or infinite set. Basically, you are saying if there is a smallest elements, then there cannot be an infinite number of predecessors to any number. Of course, this means to mean that any set without a smallest member has no lower bound, but to me that doesn't mean it's infinite. Within your theory, your proof is correct. Gee, if you have an infinite number and decrement it, it is still infinite. You can do this over and over, forever. So every infinite has an infinite predecessor, which has an infinite predecessor, etc. Nope. The predecessor of every infinite is infinite. There is no smallest infinite. The set of infinite naturals is unbounded above and below, so it would be infinite in your parlance. Nope. So, why are you talking about it? -- Smiles, Tony === Subject: Re: infinity But the smallest infinite is infinite, right? Stephen === Subject: Re: infinity Except that he doesn't believe there *is* a smallest infinite. Combined with his belief that there are a finite number of finite naturals, this produces some rather weird consequences. === Subject: Re: infinity Neither does he believe there is a largest finite, but he does believe that the largest finite is finite. If by 'weird' you mean 'contradictory', then I agree. Stephen === Subject: Re: infinity You have explained it, but I still don't see that it makes sense. I don't see, for example, how the range of the set {2,4,6,...} obtained by removing every odd natural number from the set of naturals is different from the range of the set {2,4,6,...} obtained by doubling every number in the set of natural numbers. Matt === Subject: Re: infinity Daryl McCullough said: I plan to read up on it. -- Smiles, Tony === Subject: Re: infinity Less writing and more reading would be a better plan. === Subject: Re: infinity ... Now you are left to define an infinitesimal number. Nothing of this is defined. Eh, no, most people (mathematicians at least) use things that are properly defined. You use things for which you do not give a proper definition. Except that the inverse of 0 is not defined in standard mathematics. The standard definition of inverse reads: the inverse of a number a is a number b such that a * b = 1. The inverse is noted as a^(-1), or also as 1/a. Now what is your definition of the inverse of 0? As I have not yet seen a definition of oo, this makes not yet sense. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: infinity Dik T. Winter says... I think that Tony has said something to the effect that y is infinitesimal if for all x, y*x = 0. Of course, it's inconsistent to believe that there is anything infinitesimal by that definition except 0, but that never bothered Tony. -- Daryl McCullough Ithaca, NY === Subject: Re: infinity He also defined infinitesimals as all the points that are only a finite number of points away from 0. You know, you start at 0, and go to the 'next point', e.g. .00000000...........0001 and if you only do that a finite number of times, you will have an infinitesimal. Tony believe that if you have powerful enough microscope the continuum will become discrete. Stephen === Subject: Re: infinity There are no such things as Cantorians and Non-Cantorians. There are only mathematicians and cranks, respectively. === Subject: Re: infinity This is of course part of the Cantorian party line. === Subject: Re: infinity Virgil said: My definition starts with 0 and 1. Peano's starts with 0 or 1. Those are assumed defined. Does the deifnition of an imaginary number as the square root of a negative show that any such nubmbers exist? Well, sort of yes, since there is an established operation, applied to a number that doesn't produce a real, but should rpoduce something. Likewise, we have division, and we have this number 0, and division by zero is undefined. That is, until we define it. x/oo=x*0, or x unit infinitesimals. I define it and *poof* it exists. That's math for you. So, you cannot consider ANY other perspectives on infinity? Too bad for you. Hard to do in a verbal forum. I'm getting pctures together. Where did Peano get 1 from? Where did he find the successor operator? I started with 1 and 0, and arithmetic operators, which need definition, but are assumed for the purpose of defining finiteness. That's where my numbers came from. Wave those hands, but watch out for the chandelier! Ack! My vase!! TO makes distinctions that Virgilogic can't fathom. Or in reality. The approach I am suggesting works exactly the same for finite and infinite sets defined this way. If you paid attention, you'd know that. Dedekind infinite. But that doesn't matter, you are using the element values to comapre the sets, just as I said. Without element values, you have no relative values for your bijections. In the naturals, successor means increment. Nice distraction technique. So, succession doesn't specify an order in the naturals, but succession can be used to define an order. Good distinction, there, kiddo. Fancy! Not inside standard set theory, anyway. Then there is no infinite range in which to place your infinity of unit intervals. Hence the set of finites is finite. Tada!!! Pity? When you declare the elements finite based on misuse of inductive proof, then the finiteness of your set is a direct consequence of your mistake. The Peano axioms define an infinite set. In that infinite set are infinite values which those axioms do not prohibit. It is the nonsensical prohibition of them that causes problems. I can redefine whatever I want. I just can't MAKE you agree. Not in standard set theory. The less genius, V! Can you justify any of your axioms? What I have been told many times here is that axioms need no justification. Are you changing your position on that? Like what? Your declarations are hot air. Need an anchor, Virgil? I find myself wondering what ideas Virgil has discovered on his own, and what non standard ideas he may ever have entertained. -- Smiles, Tony === Subject: Re: infinity But TO seems to get his version of real numbers out of the luminiferous aether, or at leastsome equally mythical source. But There are no infinitesimals in the standard reals from which to take reciprocals. And non-standard models like Robinson's start from the standard mode's of naturals, integers, rationals and reals that TO cannot stomach. So that TO's world of TOmatics is entirely mythical. === Subject: Re: infinity stephen@nomail.com said: What does this say, right here, below? And what does this say? You did not say that? What did you say, above? Yes, I understand, and that is generally a good definition, except when you throw finiteness into the mix. The way I understand it, a (nnonzero) finite divided by a finite is always finite, so a finite overall range of value divided by a finite range/interval divided by a finite interval/element cannot yield an infinite number of elements in that overall range. Can a finite divided by a non-zero finite be infinite? I didn't detect that question. What do you mean from? Are you asking who made them up, or how I justify them? Before I learned about set theory, I already had a concept of infinity which had to do with geometrical representations of arithmetic, hypergeometry, limits and integrals in calculus, and infinite series. So, are you asking for the source, or for the justification? There are common words whose definitions are questionable, that's for sure. The definition of a potato or pineapple is determined by nature and is not up for review until we find something that is perhaps not quite a potato or pineapple anymore. Definition abound for such concepts as love, or mind, most of which are essentially useless, but good definitions can be developed. Mathematical definitions are our own inventions, and the nature of what we are discussing depends on those definitions, as opposed to the other direction. When we are talking about abstract recursive systems that seem to have no end, there are a number of ways we can look at them. When we are talking about math in general, there are different directions we can take in our abstractions and syntheses. We can define lines using points, or points using lines. We can define quantities using sets, or sets using quantities. We can define math in terms of logic, or logic in terms of math. There is always a value in considering things from both ends when this is the case. I just started on the other end from you. That's all. I consider quantity to be what math is all about. Even logic, which some view as more basic than math, deals with quantities between 0 and 1: probabilities of true from absolutely false to absolutely true. In ways, sets would appear to be more basic. After all, formal math started by counting discrete things for accounting purposes. On the other hand, we also deal with real measures: distances, weights, speeds, volumes, which are infinitely divisible quantites, which we generally divide into finite units for the purpose of measurement, so we can count them. So, these discrete whole numbers can be viewed as specific instances of points on the real continuum. If this is the case, then perhaps it makes sense to define the real continuum and real quantities, and then state the whole numbers as a special subset of those quantities. In this case, perhaps the line should define the points, and not the other way around. Did I answer your question? Probably not. Lemme know. Then why did you ask why I have a problem with an infinite set of finite naturals? Obviously I answered that question because I do not see a contradiction, and the answer I gave was why I do not see a contradiction. It would seem not It would seem so. And? That's why I answered the question above. Just because they are an internally consistent set of rules does not mean that they are consistent with the entire body of math, or with reality. By your definitions, the set of finite naturals is infinite. By mine, it is unbounded but necessarily finite. You are willing to do away with the notion that a proper subset is not smaller than its superset for infinite sets, and willing to accept a proof that you can cut two solid balls in a finite number of pieces and reassemble them into two solid balls, each the same size as the original. I find those conclusions laughable, sorry. There are so many exceptions for the infinite case in your theory, that it seems wholly inconsistent and kludgy to me. Perhaps it's the best we can do. That seems to be the prevalent opinion within the field. I am not satisfied with that, and I don't seem to be alone. So, I have been working in the background on a better understanding of infinity, among other things, for most of my life. Sorry if that bothers your sensibilities. -- Smiles, Tony === Subject: Re: infinity Poor wording on my part. I meant you agreed that the set {1, 2, .. n } had a last element, and the set {1, 2,3 ....} did not have a last element. What does that mean? 'non finite' only makes sense when you throw 'finiteness' into the mix. It is the negation of 'finite'. There is no way to discuss 'non finite' without the concept of 'finite'. Tony, I do not care about the way you understand it. The words 'infinite' and 'finite' have well agreed upon meanings which do not lead to any contradictions. You instead have chosen your own definitions for these words? Why? Anyway, as you have been repeatedly told, the 'range' of the finite naturals is infinite according to the standard definitions. I know this contradicts your definition of 'range', but again, why you choose to use your own private definition of 'range' is beyond me. You argument seems to be, I use my own definition of range, because the standard definition contradicts my own definition. But that does not even address why you came up with your own definition in the first place. Look 105 lines up, and you will see: So you just made up a bunch of definitions for yourself, and are now claiming that the rest of the world should use your definitions, and that the rest of the world's definitions are wrong because they do not match your definitions. That answers my question You entirely missed the point. Why do you call a pineapple a pineapple? Is it an apple? Is it a pine? Does it have anything to do with pines? No. But everyone else who speaks English usually calls it a pineapple, so if you want to be understood, you call it a pineapple, despite the fact the name really does not seem to mean that. 'Infinite set' has a a precise meaning in mathematical English. Insisting that it should mean something else is the same as insisting that 'pineapple' should mean something else in my opinion. The above gobbledygook did not, but you more or less have answered my question. You just like making up your own definitions of words. And now that you have found out that your definitions contradict the standard definitions, you are upset, and want everyone to change their definitions to conform with yours. Because the ideas are essentially the same. Remember, I am talking about the standard definitions of finite, and infinite. You earlier were claiming that the standard definitions were contradictory, because you could not have an infinite set of finite elements, and seemed to be saying that is why you rejected them. Now you agree that you can have an infinite set of finite elements, according to the standard definitions. So there is no contradiction, and apparently no reason for you to reject the definitions, other than that they do not agree with your definitions. So if we decided finite meant has a last element then each element of L would be finite. I know this is not your definition of finite, nor is it the general definition of finite, but it is a pretty good definition when talking about lists. So, if we agree that finite means 'has a last element', and infinite means not finite, then L is infinite, and each member of L is finite. There is no contradiction. Again, I know these are not your definitions of finite and infinite, but they are essentially the mathematical definitions. So you agree, despite thousands of posts that seemed to indicate otherwise, that there is nothing contradictory about a 'mathematically infinite' set of 'mathematically finite' elements. Well, 'reality' does not have much to say about infinite sets, as far as I know. And the mathematical definition of 'infinite set' is perfectly consistent with the rest of mathematics (to the extent that we know mathematics is consistent). Spare me your misunderstandings of sequences and information theory. The only contradictions you ever presented were contradictions between your own definitions and the standard ones. and whatever other things you claimed. Well good luck to you. But you really need to work on your presentation, and you really need to learn how to present an argument in a logical and non-circular fashion. Stephen === Subject: Re: infinity Nor with standard math, on the evidence you preset here. === Subject: Re: infinity That's because you are looking in a mirror. === Subject: Re: infinity You are the one who brought up Achilles and time. Why did you do that if space and time are irrelevant? No, you have been repeatedly told that it is not built with infinitely many steps. People have repeatedly pointed you towards the Axiom of Infinity. A never ending set is a never ending set. Do not confuse 'exists' with physical existence. The natural numbers exist mathematically, and they are an infinite set. The standard mathematical one. You have yet to point out that contradiction. You seemed to be saying that according to the mathematical definition someone could walk an infinite distance in a finite amount of time, which is total nonsense. Stephen === Subject: Re: infinity How about something like ...98765 ? Matt === Subject: Re: infinity Robert Low said: If each post were one character longer than the last one, then that would be true. Ha! -- Smiles, Tony === Subject: Re: infinity Daryl McCullough said: Then why are you here discussing it? Hmmmm....... -- Smiles, Tony === Subject: Re: infinity Daryl, alas, is delusional. He is under the impression that a sufficiently clear explanation will get through to you. === Subject: Re: infinity Randy Poe said: N+2 No. -- Smiles, Tony === Subject: Re: infinity OK, so every finite number has the property that, when doubled, you still have a finite number. Doesn't this worry you a little? That means that the range of the doubled finite numbers is twice the range of the undoubled finite numbers. And yet, all those doubled finite numbers, even the ones past the original range, were in the original set, so the range of the doubled finite set must be the same as the range of the original set. How can a finite set have these properties, that doubling all the elements does not change the range? - Randy === Subject: Re: infinity briggs@encompasserve.org said: So, you're idea of mathematics is simple symbol manipulation? What an inspired notion! Why should math have anything to do with truth or reality? After all it's just a word game. But then, what makes anything correct or not? what's going on, and from that we abstract the quality of what's going on and abstract relationships between things, and abstract those relationships into logical relations. So, perhaps we should only operate on the logical level, having outgrown sensation, perception, emotion, memory and language? No. When you ignore the rational underpinnings of your axioms, and judge them only on whether they work as a rule in a game, then you are abstracting yourself so far that you have disconnected from the concrete reality you are abstracting from. It's really worthwhile to closely examine the justifications for each axiom that you use. -- Smiles, Tony === Subject: Re: infinity David Kastrup said: Is cardinality a number? I said you have an unclear concept of an infinite NUMBER, not set or equivalence class. 1/oo=0 | nultiply both sides by oo 1=0*oo=1 | Yay! 0*x=0 (an infinitesimal) for all finite x, not infinite x (necessarily) -- Smiles, Tony === Subject: Re: infinity But there is no real oscillation! in nowadays math. If you have SUM, it is complete and finite - that means, it does not change. Where is any oscilation? So every sum has it is value - for example 1-1+1-1+... has it - if it is 0 or 1 depends on the model (or on the length of the sum - which is natural number (for example N)). No much. I can write it down as general-number, but it is not so easy to count it with a nice formula. Definetely this sum 1/n over N = S1 is the same degree as log(N) and holds log(N) < S1 < 2*log(N). Why do you want this sum? Yes, one-compression is a way how to do summation over a set formally - if you are able to write down long enough formules. MS === Subject: Re: infinity I can't find anything on the Cornell Library web site that indicates there are restrictions on staff borrowing or access to library facilities. As far as I can tell, all staff have full privileges. http://www.library.cornell.edu/services/facultypol.html Cornell University identification cards serve as library cards. If TO has a Cornell University ID card, he's got library borrowing privileges. Methinks he has never actually darkened the doors of the library which is why he does not know if he can borrow or not. - Randy === Subject: Re: infinity May I ask you a question? Do you know, is there any possibility where to send manuscript with elaboration about infinity - I mean some Math journal - where is possibility - they will even read my work or print it? On doctor study there is one formality - that every piece of knowledge have to be printed. But in our country every Math journals are in hands of Cantorians. (I know, you are probably too, but it is a question about form - you are still open to debate.) This forum is wonderfull place, but it is probably not sufficient for formal publication. MS === Subject: Re: infinity !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ Look, the problem with the Cantorians is the same as with the Einsteinians in physics and the Wittgensteinians in philosophy. You can't pretend those people did not live and assume that their insights have somehow disappeared again. The naiver world views they shattered are no longer tenable. Because their bases have been shown to be hollow. If you want to return to older ventures, take up painting or music. While you can't undo Bach's life, music in Baroque and Renaissance style still makes sense even after him. It sounds like you should rather aim for philosophy journals. I am afraid that from what I've read here from you that you'll have problems placing your paper with a competent journal of any discipline, though. I am not a mathematician myself, but an engineer. I have seen my share of pure bull papers published in conferences and journals, of the kind if we throw even more research money and computer power down the drain, the fundamental information theoretic barriers we don't seem to be able to overcome will magically go away. Even there, people not only want to believe themselves that they can prevail against the _proven_ impossible, but they managed to publish and even get grant money for doing this. For a while. So I am not saying that it would be impossible for you to finish a doctorate degree on this basis. But it seems unlikely that it would make you a mathematician, so I am not sure that it would not, in the long run, be a step forward into a dead end. You bet. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: infinity offer a piece of advice. Your use of the term Cantorian makes you essentially appear as a crank. The journals are not in the hands of Cantorians...they are in the hands of Mathematicians. Only cranks speak of Georg Cantor as if he is offering some contraversial propositions. It is similar to someone wishing to publish in Physics to disprove Relativity, as if it is still up for debate. Now, you can certainly investigate alternative frameworks. For example, if you wished to start off with ZFC and remove the Axiom of Infinity to replace it with something else to see what happens, that is certainly fine. There are books on Set Theory already that discuss alternatives to the Axiom of Choice, the Axiom of Foundation, the inclusion of other axioms, etc. If you do go down this round, make sure you are using mathematical rigor, not personal philosophical or emotional leanings, to direct you. Jonathan Hoyle === Subject: Re: infinity There are four possible combinations of truth values for finite(x) and infinite(x): 1. x is finite and x is infinite 2. x is finite and x is not infinite 3. x is infinite and x is not finite 4. x is not infinite and x is not finite Your set of statements is consistent with case (1) (x is finite and infinite), with case (2) as Daryl proved above, with case (3) when x is outside the interval [0,1], and with case (4) when x = 0. So it seems that 0 is neither finite nor infinite, and that it is possible for numbers to be finite and infinite simultaneously -- that is, that one can't necessarily decide whether a number is finite, infinite, both, or neither. But that's precisely what a definition is supposed to allow one to do. Thus, it appears that you do not have a definition. Matt === Subject: Re: infinity William Hughes said: Okay, then that is also one less than the last element of your set: {1,2,3,.....,R-1,R, R+1}. (R+1)-1=R, the largest difference in the set. {2,4,6,...,R-1,R+1} Okay Yes {2,4,6,...,2R,2R+2} Is 2R in B? Nope. -- Smiles, Tony === Subject: Re: infinity No. A does not have a last element. No, since A does not have a last element, B does not have a last element. No, since A does not have a last element. C does not have a last element. If R is a natural number, then 2R is an even natural number. Since B contains all even natural numbers then 2R is in B. -William Hughes === Subject: Re: infinity Yes, A has no last element, it is true, but SUMMATION over A has last element and this is SUM {1} over A =: R. Then holds, that A1 = {2,3,...} has itĒs summation R-1 and the set A{1,2,3} has itĒs summation R-3. No, it is not true. A has not itĒs last element, but B is set of all SUMMATIONS over set A! And it have last element! Every set has it, even very large unreachable ordinals have. The same as before... This depends on the way you are making B. You have A ={1,2,3,...}, B = {2,4,6,...}. If You ask, how big is B intersection A, then the answer is |A|/2. If |B| = |A| = N, then summation of all members of A Sum n in A = N*(N-1)/2 and Sum n in B = N*(N-1). MS === Subject: Re: infinity No it doesn't. B contains all even naturals (the question of the size of B is not important here). If R is a natural number then 2R is an even natural number and since B contains all even naturals numbers , B contains 2R. Above you claim that R is a SUMMATION. If a SUMMATION is a natural number then B contains 2R. If a SUMMATION is not a natural number then B does not contain 2R (B only contains natural numbers). On the other hand if R is a pink grapefruit then B does not contain 2R either. Note, according to TO, R is a natural number. -William Hughes === Subject: Re: infinity According to this set of statements, 0 is neither finite nor infinite. Is that what you intended? Matt === Subject: Re: infinity Matt Gutting says... Actually, it is *consistent* with those axioms to assume that 0 is neither finite nor infinite. It is also consistent to assume that it is *both* infinite and finite. It is consistent to assume that *every* element is infinite. It is consistent to assume that every element is *finite*. That's why it isn't really a definition. It doesn't allow you to prove that any particular thing is infinite or not infinite. -- Daryl McCullough Ithaca, NY === Subject: Re: infinity Actually I went through that in another post, but I may have got my logic wrong. I was looking to see whether the concepts of 'finite' and 'infinite' presented were mutually exclusive and exhaustive; I discovered they were neither. Matt === Subject: Re: infinity Torkel Franzen said: What DOES it mean to you, Torkel? -- Smiles, Tony === Subject: Re: infinity Fnoffleboffle. Anything else is Cantorian fog. === Subject: Re: infinity Does one pronounce Fnoffleboffle with the accent on the first syllable or on the last, or does one not pronounce it at all? === Subject: Re: infinity Yes. Matt === Subject: sets I have 3 questions about sets: 1. how can a set on natural numbers be countable, when it is infinite? 2. what is a cardinal number and how does it differ from all other numbers? 3. what are aleph0 and aleph1? === Subject: Re: sets I'll give answers in less technical terms, in case the original poster didn't follow all the jargon... A set is countable if every member of that set can be assigned a natural number uniquely. This is a one-to-one mapping between that set and the set of naturals. This is a way of counting every member of the set. Since the set of natural numbers is infinite, any set that can be mapped to the set of naturals is also infinite, and countable. There are also sets that are infinite but uncountable, which are larger than the set of naturals. Cardinality is the size or measure of a set. If a set is finite, its cardinality will also be a natural number. If the set is infinite, its size will be a transfinite (infinite) cardinal number, which is not a natural number. Aleph_0 is the first transfinite (infinite) cardinal number, and Aleph_1 is the second. Aleph_0 is the size of the set of natural numbers. The Continuum Hypothesis states that Aleph_1 = 2^Aleph_0, but this cannot be proved or disproved in standard set theory. http://en.wikipedia.org/wiki/Cardinal_number http://en.wikipedia.org/wiki/Transfinite http://en.wikipedia.org/wiki/Continuum_hypothesis === Subject: Re: sets Try http://mathworld.wolfram.com/CardinalNumber.html and in future, learn to use Google. === Subject: Re: sets Since an answer was given to questions 1 and 3 I will take a stab at number 2. A cardinal is an ordinal that has no bijection with any ordinal preceeding it. An ordinal is a special represantative of a totaly ordered set. You can read more about this in many textbooks. Cardinals and numbers are quite different, if you treat numbers as abstract concepts. If you take the definition that a number is the set of all smaller (positive) numbers, then any such number is actually a cardinal. However there are a lot of infinite cardinals. A lot of the arithmetic of ordinary numbers can be extended to cardinals and there are whole books on cardinal arithmetic. The situation turns out to be not so trivial and quite a few results depend on the axiom of choice. I hope this helps. === Subject: Re: sets days. My association with the Department is that of an alumnus. What is a set on natural numbers? Do you mean the set of all natural numbers? The definition of countable is: either finite, or else may be put in one-to-one corerspondence with the natural numbers. Since the identity function clearly puts the natural numbers in one-to one correspondence with the natural numbers, that makes the set of all natural numbers countable (though infinite). An alternative definition might be that a set is countable if and only if it can be put in one-to-one correspondence with a subset of the natural numbers, in which case it is obvious that any subset of the natural numbers (the other possible interpretation of your phrase a set on natural numbers) is countable. A cardinal number is a special kind of ordinal number. They differ from other numbers in many ways, just as othe rnumbers differ form each other in many ways: integers differ from natural numbers, rationals from integers, reals from rationals, complex from reals, etc. Cardinal numbers are intuitively related to the size (in the sense of cardinality, which may not agree with usual intuition) of sets, generalizing in some ways the usual finite size numbers (the natural numbers), but including infinite sets as well. In any case, you might want to try starting at: http://en.wikipedia.org/wiki/Cardinal_number Aleph0 is the cardinal number that corresponds to the natural numbers; intuitively, if cardinals are a measure of size, it represents the smallest a set can be and still be infinite. Aleph1 is the smallest cardinal strictly larger than aleph 0. If cardinals are a measure of size, then aleph 1 represents the smallest a set can be and not be countable. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: sets It's a matter of definition. A set X is said to be countable if there is an injective mapping (a mapping that is one-to-one) from X into the natural numbers. This means that each member of X is associated with some natural number, and therefore X is countable. There is more than one way to define a cardinal number. Rather than focusing on the actual definition, which is a bit arcane, it's best to understand what it means for two sets to have the same cardinality. This simply means that there is a bijective mapping (a mapping that is one-to-one and onto) between them. Aleph_0 is the smallest infinite cardinal. Any set that can be mapped bijectively to the natural numbers is said to have cardinality aleph_0. Some examples are the integers, the rationals, and the algebraic numbers. Aleph_1 is the smallest uncountable ordinal. It's also the cardinality of the set of all countable ordinals. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: sets I'll take a stab... Its the (most accepted) definition of countable: either a set is finite OR there exists a bijection on N. So this may force you to abandon your old ideas of what countability means (in the colloquial sense). Those refer to the cardinality of N and the power set of N, respectively. Jason === Subject: Re: sets No: aleph1 is the next infinite cardinal up after aleph0. The continuum hypothesis says that it is the same as the cardinality of the power set of N (which is the same as the cardinality of R). === Subject: Re: sets days. My association with the Department is that of an alumnus. The first one is correct; the second assertion is the Continuum Hypothesis. Aleph 1 is the smallest cardinal strictly larger than Aleph0. The cardinality of the power set of N is usually denoted c, for continuum (same cardinality as the real numbers, or real line). The statement that c=Aleph1 is the Continuum Hypothesis, which is undecidable in ZFC. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: the REVELATIONS of a CRANK as a reply to WILESproof of FLT I am sorry.I apologized.I beleived that the bet refers to any body claiming the same thing. Gheorge Ghiata === Subject: Classical ruin problem I am trying to learn and solve classical ruin problem on my own. I am currently reading Feller's Introduction to probability theory and it's applications Vol.1. However I find it difficult to follow him as he has skipped a few middle steps. These steps are not obvious to me and I would appreciate if someone would help me with it. The problem is the classic ruin problem with 2-boundary absorption (pp. 344, 3rd Edition). Let Q be ultimate ruin, P - Proabability of winning. q - probability of losing 1 hand, p - probability of winning 1 hand. The two absorbing boundaries are 0 (lost everything) or A (gained everything the other gambler had). Let 'z' the variable that denotes the stages in between 0 and A. After the first trial the gambler's fortune is z-1 (if he loses) or z+1 (if he wins that hand) when z = 1, Q(0) = 1 (absolute certainty), therefore Q(1) = p.Q(2) + q when z = a-1, Q(a) = 0, therefore Q(a-1) = q.Q(a-2) Q(0) = 1 and Q(a) = 0 are the boundary conditions. provide the solution as Q(z) = A + B(q/p)^z and Q(z) = (q/p)^z I am struggling to see how he finds these particular solutions. I am planning on reading this chapter further but I cannot proceed without understanding this part. Any suggestions, hints or explanations that will help clarify things for me are welcome. If you know what I should read up on to gain an understanding about this method, kindly let me know. === Subject: Re: Classical ruin problem A standard method is to ASSUME a solution of the form r^z (r to the power z), and to use the recurrence equation to find r. In this case we must have r^z = p.r^(z+1) + q.r^(z-1). Dividing by r^z we have 1 = pr + q/r, which has two solutions: r = 1 or r = q/p. If you multiply through by r, you get a quadratic equation with those two roots; but, you can also see directly that they both work. In general, for a linear recurrence equation of the form f(n) = a_1 f(n-1) + a_2 f(n-2) + ... + a_k f(n-k) (without specified boundary conditions), the solution will be of the form f(n) = r^n, where r satisfies the equation obtained by substituting this form into the 100 years, but I don't know who first realized it. (It is standard material in Linear Systems courses, etc.) Feller will make considerable use of this technique in later chapters, so learning it now is useful to you. Good luck. Ray Vickson Adjunct Professor, University of Waterloo === Subject: Cylinder plane intersection Could anyone point the way to determining the equation of the ellipse resulting from the intersection of the plane and the cylinder. More specifically this problem is a 3 dimensional problem. In other words unlike solving the equation of two 2-dimentional objects say a circle and a straight line to obtain a point or a pair of intersecting points I have to find the locus of intersecting points. I've thought along the lines of parametrizing the equation and assuming a solution etc but to no success. Arjun PS: I realise that the plane and cylinder might not even intersect or intersect to produce a tangent line or a pair of lines. === Subject: Re: 0.999... = 1? (I know, a beaten dead horse) who didn't tell them not to, then -- Sargent Pepper (the producer) ?? --Trickier Dick's Obnoxico? http://larouchepub.com/other/2005/3237energy_heist.html