mm-4589 === Subject: Re: Differentiabilty and Lipschitz condition, is this true? > We know that, if f:I--> R is differentiable on the > open interval I, then I contains a subinterval where > f is Lipschitz. But I think we don't need full > differentiabilty on I. It suffices that all of the > Dini's derivatives exist everywhere on I. Suppose there's no subinterval of I where f is > Lipschitz. Then, we can find a_1 < b_1 in I such that b_1 - a_1 < 2^(-1) >|f(b_1) - f(a_1)| >= 2^1 (b_1 - a_1) By assumption, there's no subinterval of (a_1 , b_1) > where f is Lipschitz. So, we can apply the same > reasoning to (a_1, b_1) to get a < a_1 < a_2 < b_2 < > b_1 < b in a similar way. By means of a simple > inductive process, we get sequences a_n and b_n in > (a, b) such that, for each n, a < a_1....< a_n < b_n...< b_1 < b >b_n - a_n < 2^(-n) >|f(b_n) - f(a_n)| >= 2^n (b_n - a_n) (1) So, [a_n, b_n] is a nested sequence of closed > subintervals of (a, b) whose length approaches 0. It > follows there exists one, and only one, x common to > all of the intervals [a_n , b_n]. We have x = > supremum a_n = infimum b_n and x in (a, b). For any h > 0 such that (x - h, x + h) is in (a, b), > there are infinitely many a_n in (x - h, x) and > infinitely many b_n in (x , x + h). For each n, it > follows from (1) that |f(b_n) - f(a_n)|/(b_n - a_n) >= 2^n Hence, this Newton quotient is unbounded for any of > such h's, which shows f is not differentiable at x. > In addition, since |f(b_n) - f(a_n)|/(b_n - a_n) <= |f(b_n) - > f(x)|/(b_n - a_n) + |f(x) - f(a_n)|/(b_n - a_n) and |f(b_n) - f(x)|/(b_n - a_n) < |f(b_n) - f(x)|/(b_n - > x) >|f(x) - f(a_n)|/(b_n - a_n) < |f(x) - f(a_n)|/(x - > a_n) It follows that |f(b_n) - f(a_n)|/(b_n - a_n) <= |f(b_n) - > f(x)|/(b_n - x)+ |f(x) - f(a_n)|/(x - a_n) Hence, if the Dini's derivative exists at x+ (or > x-), then the Newton's quotient must get unbounded at > x- (or respectively at x+). We see it's impossible > that both f'(x-) and f'(x+) exist simultaneously. We conclude that, if all the Dini's derivatives > exist everywhere on I, then I contains a subinterval > where f is Lipschitz. Is this OK? > > What part seems questionable? > Artur > > David C. Ullrich Ok, thank you! Artur === Subject: Differentiabilty and Lipschitz condition, is this true? We know that, if f:I--> R is differentiable on the open interval I, then I contains a subinterval where f is Lipschitz. But I think we don't need full differentiabilty on I. It suffices that all of the Dini's derivatives exist everywhere on I. Suppose there's no subinterval of I where f is Lipschitz. Then, we can find a_1 < b_1 in I such that b_1 - a_1 < 2^(-1) |f(b_1) - f(a_1)| >= 2^1 (b_1 - a_1) By assumption, there's no subinterval of (a_1 , b_1) where f is Lipschitz. So, we can apply the same reasoning to (a_1, b_1) to get a < a_1 < a_2 < b_2 < b_1 < b in a similar way. By means of a simple inductive process, we get sequences a_n and b_n in (a, b) such that, for each n, a < a_1....< a_n < b_n...< b_1 < b b_n - a_n < 2^(-n) |f(b_n) - f(a_n)| >= 2^n (b_n - a_n) (1) So, [a_n, b_n] is a nested sequence of closed subintervals of (a, b) whose length approaches 0. It follows there exists one, and only one, x common to all of the intervals [a_n , b_n]. We have x = supremum a_n = infimum b_n and x in (a, b). For any h > 0 such that (x - h, x + h) is in (a, b), there are infinitely many a_n in (x - h, x) and infinitely many b_n in (x , x + h). For each n, it follows from (1) that |f(b_n) - f(a_n)|/(b_n - a_n) >= 2^n Hence, this Newton quotient is unbounded for any of such h's, which shows f is not differentiable at x. In addition, since |f(b_n) - f(a_n)|/(b_n - a_n) <= |f(b_n) - f(x)|/(b_n - a_n) + |f(x) - f(a_n)|/(b_n - a_n) and |f(b_n) - f(x)|/(b_n - a_n) < |f(b_n) - f(x)|/(b_n - x) |f(x) - f(a_n)|/(b_n - a_n) < |f(x) - f(a_n)|/(x - a_n) It follows that |f(b_n) - f(a_n)|/(b_n - a_n) <= |f(b_n) - f(x)|/(b_n - x)+ |f(x) - f(a_n)|/(x - a_n) Hence, if the Dini's derivative exists at x+ (or x-), then the Newton's quotient must get unbounded at x- (or respectively at x+). We see it's impossible that both f'(x-) and f'(x+) exist simultaneously. We conclude that, if all the Dini's derivatives exist everywhere on I, then I contains a subinterval where f is Lipschitz. Is this OK? Artur === Subject: What would a fair proof for this claim be Some guys, good at number theory, were discussing a proof for the following claim: Let p_n, n=1,2,3... be the sequence of prime numbers. For every k >1, we have p_n < n^k for infinitely many indexes n. They were using log approximation of prime numbers. But, I found another somewhat silly argument: If, for some k > 1, the claim is not true, then there exists m such that p_n >= n^k for every n > m, which is the same as 0 < 1/p_n <= 1/(n^k), n > m Since k >1, Sum 1/(n^k) converges, so implying, by comparison, that so does Sum 1/p_n. But since this series, according to a well known result, in fact diverges, we get a contradiction that shows the claim is true. Actually, this claim has very little to do with prime numbers, it's immediate it's true for any sequence a_n of positive numbers such that Sum 1/a_n diverges. These were the only properties of p_n this proof took into account. So, according to the guys, all the beauty and all the charm of the problem were gone. My proof destroyed them. The only thing I can think of to say in favor of this proof is that it's indeed a proof. Though not very elegant. How would an elegant proof, able to save the problem, go? (The problem with this problem is that it was mistakenly stated in terms of prime numbers. It might be stated as Show that, if a_n is a sequence of positive numbers such that Sum 1/a_n diverges, then, for every k > 1, we have a_n < n^k for infinitely many n) Sharon === Subject: Re: What would a fair proof for this claim be posting-account=bSICGQkAAADSbkxAJ5uMxFegr4rp0Qig Gecko/20071115 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) > Some guys, good at number theory, were discussing a proof for the following claim: Let p_n, n=1,2,3... be the sequence of prime numbers. For every k >1, we have p_n < n^k for infinitely many indexes n. They were using log approximation of prime numbers. But, I found another somewhat silly argument: If, for some k > 1, the claim is not true, then there exists m such that p_n >= n^k for every n > m, which is the same as 0 < 1/p_n <= 1/(n^k), n > m Since k >1, Sum 1/(n^k) converges, so implying, by comparison, that so does Sum 1/p_n. But since this series, according to a well known result, in fact diverges, we get a contradiction that shows the claim is true. Actually, this claim has very little to do with prime numbers, it's immediate it's true for any sequence a_n of positive numbers such that Sum 1/a_n diverges. These were the only properties of p_n this proof took into account. So, according to the guys, all the beauty and all the charm of the problem were gone. My proof destroyed them. The only thing I can think of to say in favor of this proof is that it's indeed a proof. Though not very elegant. How would an elegant proof, able to save the problem, go? I guess elegance is more subjective than I would have thought. I find your proof very elegant, and I like the way it ties together two different areas of mathematics. And actually I suspect that the proof of the divergence of sum(1/p) may involve something similar anyway. - Randy === Subject: Re: What would a fair proof for this claim be > On Apr 10, 1:24 pm, Sharon proof for the following claim: Let p_n, n=1,2,3... be the sequence of prime > numbers. For every k >1, we have p_n < n^k for infinitely many indexes n. They were using log approximation of prime numbers. > But, I found another somewhat silly argument: If, for some k > 1, the claim is not true, then > there exists m such that p_n >= n^k for every n > m, which is the same as 0 < 1/p_n <= 1/(n^k), n > m Since k >1, Sum 1/(n^k) converges, so implying, by > comparison, that so does Sum 1/p_n. But since this > series, according to a well known result, in fact > diverges, we get a contradiction that shows the claim > is true. Actually, this claim has very little to do with > prime numbers, it's immediate it's true for any > sequence a_n of positive numbers such that Sum 1/a_n > diverges. These were the only properties of p_n this > proof took into account. So, according to the guys, > all the beauty and all the charm of the problem were > gone. My proof destroyed them. The only thing I can think of to say in favor of > this proof is that it's indeed a proof. Though not > very elegant. How would an elegant proof, able to save the > problem, go? > > I guess elegance is more subjective than I would have > thought. > > I find your proof very elegant, and I like the way > it ties together two different areas of mathematics. > > And actually I suspect that the proof of the > divergence > of sum(1/p) may involve something similar anyway. > > - Randy There's a proof based on Euler product. It's not not difficult. Sharon === Subject: Re: What would a fair proof for this claim be > Some guys, good at number theory, were discussing a proof for the following > claim: Let p_n, n=1,2,3... be the sequence of prime numbers. For every k >1, we > have p_n < n^k for infinitely many indexes n. They were using log approximation of prime numbers. But, I found another > somewhat silly argument: If, for some k > 1, the claim is not true, then there exists m such that p_n >= n^k for every n > m, which is the same as 0 < 1/p_n <= 1/(n^k), n > m Since k >1, Sum 1/(n^k) converges, so implying, by comparison, that so does > Sum 1/p_n. But since this series, according to a well known result, in fact > diverges, we get a contradiction that shows the claim is true. Actually, this claim has very little to do with prime numbers, it's > immediate it's true for any sequence a_n of positive numbers such that Sum > 1/a_n diverges. These were the only properties of p_n this proof took into > account. So, according to the guys, all the beauty and all the charm of the > problem were gone. My proof destroyed them. The only thing I can think of to say in favor of this proof is that it's > indeed a proof. Though not very elegant. How would an elegant proof, able to save the problem, go? > > I guess elegance is more subjective than I would have > thought. > > I find your proof very elegant, and I like the way > it ties together two different areas of mathematics. > > And actually I suspect that the proof of the divergence > of sum(1/p) may involve something similar anyway. The divergence of sum (1 / p) is Theorem 1.13 in Apostol, Introduction to Analytic Number Theory, and the proof given there uses only the divergence, for all integers Q, of sum ( 1 + n Q )^(-1). Apostol cites James A Clarkson, On the series of prime reciprocals, Proc Amer Math Soc 17 (1966) 541, MR 32 #5573. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Witt vectors and DVR posting-account=-qlJtAkAAADgsMLtrN9n4qex_kHnFTZO Gecko/20080325 Ubuntu/7.10 (gutsy) Firefox/2.0.0.13,gzip(gfe),gzip(gfe) DVR that is the completion of W(k)(x_1,...x_d), where W(k) are the Witt vectors associated to the perfect field k. What does he mean? What ring is W(k)(x_1,...x_d)? The only reasonable answer for me is that W(k)(x_1,...x_d) is the ring of the formal Lauren series in d variables with coefficient in W(k). Note that W(k) (x_1,...x_d) cannot be the field of fraction of W(k)[x_1,...x_d] because otherwise it does not make sense speak about it completion. === Subject: Integration / Estimating Question posting-account=kDJ5tQoAAACHGG2HeH5kSmDxVb73tjc7 Gecko/20080311 Firefox/2.0.0.13,gzip(gfe),gzip(gfe) I took a calc exam last night and had a problem on it that has me stumped. I've been out of school for many years and am going back to finally finish up so even my basic math skills are rusty... the question was: Estimate the area under the graph of f(x) = 2 - x^(1/3) between -2 and 2 using the midpoint rule, with 2 rectangles of equal length My problem is, when i take a cubed root of a negative number i get an imaginary number, which i cannot graph. So i figured that I would work the problem from 0 to 2, making the midpoints 1/2 and 3/2. After working the problem that way, i came up with an estimate of just over 2. When i turned the exam in, I asked the professor about that question and he said my solution was wrong but didn't say why other than you can take a root of a negative number. I tried solving the problem as an integral on my calculator [-2,2] and get a crazy answer with + i. I'm assuming the 'i' stands for imaginary? Anyways, can someone shed some light on this? I'm dying to know how to solve the problem and won't have the graded exam back for at least a week... === Subject: Re: Integration / Estimating Question posting-account=G_G-iQoAAAB08LNQidt_LsMkopmIb4ZS Gecko/20060111 Firefox/1.5.0.1 Mnenhy/0.7.3.0,gzip(gfe),gzip(gfe) > I took a calc exam last night and had a problem on it that has me > stumped. I've been out of school for many years and am going back to > finally finish up so even my basic math skills are rusty... the question was: Estimate the area under the graph of f(x) = 2 - x^(1/3) between -2 and > 2 using the midpoint rule, with 2 rectangles of equal length My problem is, when i take a cubed root of a negative number i get an > imaginary number, which i cannot graph. So i figured that I would work > the problem from 0 to 2, making the midpoints 1/2 and 3/2. After working the problem that way, i came up with an estimate of just > over 2. When i turned the exam in, I asked the professor about that > question and he said my solution was wrong but didn't say why other > than you can take a root of a negative number. I tried solving the > problem as an integral on my calculator [-2,2] and get a crazy answer > with + i. I'm assuming the 'i' stands for imaginary? Anyways, can someone shed some light on this? I'm dying to know how to > solve the problem and won't have the graded exam back for at least a > week... > What is the cube of ( -1 ) ? Bill J === Subject: Re: Integration / Estimating Question posting-account=kDJ5tQoAAACHGG2HeH5kSmDxVb73tjc7 Gecko/20080311 Firefox/2.0.0.13,gzip(gfe),gzip(gfe) I took a calc exam last night and had a problem on it that has me > stumped. I've been out of school for many years and am going back to > finally finish up so even my basic math skills are rusty... the question was: Estimate the area under the graph of f(x) = 2 - x^(1/3) between -2 and > 2 using the midpoint rule, with 2 rectangles of equal length My problem is, when i take a cubed root of a negative number i get an > imaginary number, which i cannot graph. So i figured that I would work > the problem from 0 to 2, making the midpoints 1/2 and 3/2. After working the problem that way, i came up with an estimate of just > over 2. When i turned the exam in, I asked the professor about that > question and he said my solution was wrong but didn't say why other > than you can take a root of a negative number. I tried solving the > problem as an integral on my calculator [-2,2] and get a crazy answer > with + i. I'm assuming the 'i' stands for imaginary? Anyways, can someone shed some light on this? I'm dying to know how to > solve the problem and won't have the graded exam back for at least a > week... > What is the cube of ( -1 ) ? Bill J would be -1 === Subject: Re: Integration / Estimating Question posting-account=bSICGQkAAADSbkxAJ5uMxFegr4rp0Qig Gecko/20071115 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) > I took a calc exam last night and had a problem on it that has me > stumped. I've been out of school for many years and am going back to > finally finish up so even my basic math skills are rusty... the question was: Estimate the area under the graph of f(x) = 2 - x^(1/3) between -2 and > 2 using the midpoint rule, with 2 rectangles of equal length My problem is, when i take a cubed root of a negative number i get an > imaginary number, which i cannot graph. So i figured that I would work > the problem from 0 to 2, making the midpoints 1/2 and 3/2. After working the problem that way, i came up with an estimate of just > over 2. When i turned the exam in, I asked the professor about that > question and he said my solution was wrong but didn't say why other > than you can take a root of a negative number. I tried solving the > problem as an integral on my calculator [-2,2] and get a crazy answer > with + i. I'm assuming the 'i' stands for imaginary? Anyways, can someone shed some light on this? I'm dying to know how to > solve the problem and won't have the graded exam back for at least a > week... > What is the cube of ( -1 ) ? Bill J would be -1 OK, so (-1)^3 = -1. Now take the cube root of both sides of that equation. Are you sure the cube root of -1 is imaginary? - Randy === Subject: Re: Integration / Estimating Question posting-account=kDJ5tQoAAACHGG2HeH5kSmDxVb73tjc7 Gecko/20080311 Firefox/2.0.0.13,gzip(gfe),gzip(gfe) I took a calc exam last night and had a problem on it that has me > stumped. I've been out of school for many years and am going back to > finally finish up so even my basic math skills are rusty... the question was: Estimate the area under the graph of f(x) = 2 - x^(1/3) between -2 and > 2 using the midpoint rule, with 2 rectangles of equal length My problem is, when i take a cubed root of a negative number i get an > imaginary number, which i cannot graph. So i figured that I would work > the problem from 0 to 2, making the midpoints 1/2 and 3/2. After working the problem that way, i came up with an estimate of just > over 2. When i turned the exam in, I asked the professor about that > question and he said my solution was wrong but didn't say why other > than you can take a root of a negative number. I tried solving the > problem as an integral on my calculator [-2,2] and get a crazy answer > with + i. I'm assuming the 'i' stands for imaginary? Anyways, can someone shed some light on this? I'm dying to know how to > solve the problem and won't have the graded exam back for at least a > week... > What is the cube of ( -1 ) ? Bill J would be -1 OK, so (-1)^3 = -1. Now take the cube root of both sides of that equation. Are you sure the cube root of -1 is imaginary? - Randy Actually, i'm finding out that i am unsure of many things math related... I came to that conclusion when i entered root(-1,3) in my TI-89 and it returned 'i'. Hah, ok... -1^(1/3) = -1 because (-1) * (-1) * (-1) = -1. wow, i wish i could delete this entire thread now... i'm feeing pretty stupid... === Subject: Re: Integration / Estimating Question posting-account=bSICGQkAAADSbkxAJ5uMxFegr4rp0Qig Gecko/20071115 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) > I took a calc exam last night and had a problem on it that has me > stumped. I've been out of school for many years and am going back to > finally finish up so even my basic math skills are rusty... the question was: Estimate the area under the graph of f(x) = 2 - x^(1/3) between -2 and > 2 using the midpoint rule, with 2 rectangles of equal length My problem is, when i take a cubed root of a negative number i get an > imaginary number, which i cannot graph. So i figured that I would work > the problem from 0 to 2, making the midpoints 1/2 and 3/2. After working the problem that way, i came up with an estimate of just > over 2. When i turned the exam in, I asked the professor about that > question and he said my solution was wrong but didn't say why other > than you can take a root of a negative number. I tried solving the > problem as an integral on my calculator [-2,2] and get a crazy answer > with + i. I'm assuming the 'i' stands for imaginary? Anyways, can someone shed some light on this? I'm dying to know how to > solve the problem and won't have the graded exam back for at least a > week... > What is the cube of ( -1 ) ? Bill J would be -1 OK, so (-1)^3 = -1. Now take the cube root of both sides of that equation. Are you sure the cube root of -1 is imaginary? - Randy Actually, i'm finding out that i am unsure of many things math > related... I came to that conclusion when i entered root(-1,3) in my > TI-89 and it returned 'i'. Every number has 3 cube roots, and I could see a calculator not knowing whether it was supposed to return a complex root by default. However, i is not one of the cube roots of -1. i^3 = i^2*i = (-1)*i = -i The complex cube roots of -1 are (1 +- sqrt(3))/2 - Randy === Subject: Re: Is Heraclitian (aka Calvinball) Chess possible? posting-account=lYey8QoAAACmIFs3A89MSlRonUPe3wtR Gecko/20080311 Firefox/2.0.0.13,gzip(gfe),gzip(gfe) On Apr 8, 10:19 pm, Wlodzimierz Holsztynski (Wlod) > On Apr 8, 6:40 pm, Wlodzimierz Holsztynski (Wlod) So the basic framework for the ultimate chess variant would be: Can > you have a framework for chess and variants that would enable a person > to NEVER play chess the same way twice (by the exact same set of > rules)? It's only to easy. > I was very conservative. In fact, I have many more > of them, and each sequence consists of astronomically > many variants. (Variants from different sequences > are always different, and so are any two from any > given sequence). Astronomically large isn't infinite though. You can see one version laid out by George Duke, in 91 1/2 Trillion Falcon Chess variants, to see the boundaries here: http://www.chessvariants.org/index/msdisplay.php?itemid=MSninety-oneanda The number studied has gotten larger than 91 1/2 Trillion by the way. However, it still isn't unbound or infinite. Perhaps someone mathematically can show the number of potential rules governing any system is finite in nature, then Heraclitian (and its Calvinball version) wouldn't be possible. - Rich === Subject: Re: Is the empty set a number? Numbers are a type of naming in relation to things. An empty set can only make sense in relation to other sets with a definable property. The empty set designated as '0' is 'even' in relation to the integer number line pointing to size i.e., -2,-1, 0, 1, 2, etc... But I think what your question is whether an empty set can be considered a 'thing.' Numbers are not things. None of the numbers are things. They are just abstract concepts to rigidly classify and name things in rigid order. When you name a group of rocks '3.' The rocks are not '3', but the rigid naming relying on the abstract concept of '3' is '3.' Abstract concepts like numbers cannot exist alone. Despite being 'nouns', they must also act as 'predicates' in that they must hook onto something in relation to something else to make sense. This is because they are abstract. While all predicates are abstract not all nouns are. So it is important to make the splice in nouns. This 'hooking' can take place among other 'abstract objects' and/or evemtually relationshps to physical things. Otherwise, you end up with nasty metaphysical dangling participles like Plato. I hope that I have adequately answered your question. B.T. === Subject: Re: Is the empty set a number? <164014.1207855023387.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW SV1),gzip(gfe),gzip(gfe) > Numbers are a type of naming in relation to things. I agree > An empty set can only make sense in relation to other sets with a definable property. Well here i am not sure i agree, in my mind the set only is a container of a group and have no value on its own. At least not a value that have numerical significance in relation to the rational numbers. In the same way color is not a member in of the group red, green, blue, yellow it is only a definition in this case of the fact that colors exist. Color in itself is nothing without the use of a member. The expression number is the full set of all numbers, but the empty set lack numbers. The empty set is only an abstract notion of the fact we can group things in brackets or containers. I am not sure i consider the set in itself a mathematical entity only an abstract idea that we can define a group without members. (i am pretty sure the set dont exist without members). The abstract idea of no group exist though. It does not make it a mathematical entity. > The empty set designated as '0' is 'even' in relation to the integer number line pointing to size i.e., -2,-1, 0, 1, 2, etc... > I do not consider 0 to be a number i consider it to be a human invention to express the fact that the group have no members. > But I think what your question is whether an empty set can be considered a 'thing.' No i consider the empty set to be a construct of human mind to express absense, i am sure THE EMPTY SET is not a thing, and i doubt that it is a NECESSARY mathematical entity. The way i see it the human mind created logic and math to describe grouprelations, without groups there would be no members no sets and no math. Without groups there can not be THINGS but as soon you have a member or a group YOU SURELY HAVE THINGS. zero concept was developed. > Numbers are not things. None of the numbers are things. They are just abstract concepts to rigidly classify and name things in rigid order. When you name a group of rocks '3.' The rocks are not '3', but the rigid naming relying on the abstract concept of '3' is '3.' Sure numbers are things, they are members of NUMBERS and have the GROUP property used as proof of their thingyness. 0 is not a member of numbers though. > Abstract concepts like numbers cannot exist alone. Despite being 'nouns', they must also act as 'predicates' in that they must hook onto something in relation to something else to make sense. True is that they would be meaningless in a reality without things. But if abstract intelligence can exist, the idea of NUMBERS can exist once the concept is developed. So i am not against the idea of abstract numbers itself only that 0 and empty set is members of NUMBERS. > This is because they are abstract. While all predicates are abstract not all nouns are. So it is important to make the splice in nouns. > This 'hooking' can take place among other 'abstract objects' and/or evemtually relationshps to physical things. Otherwise, you end up with nasty metaphysical dangling participles like Plato. I hope that I have adequately answered your question. B.T. JT === Subject: Re: Is the empty set a number? > On Apr 7, 12:24 pm, jonas.thornv...@hotmail.com On 7 Apr, 18:08, G.E. Ivey Do you understand the concept of equivalence? > æone way of defining the natural numbers is to set > 0 to be the empty set, {}, 1 to be the set whose > only member is the empty set, {{}}, 2 to be the set > whose only members are the empty set and {{}}- that > is, whose only members are 0 and 1- {0, 1}, etc. æ æWe can then define the successor of any > number, x, to be the set containing x and all of its > members and show that Peano's axioms for the natural > numbers hold. > We could define addition of two such things by > x+ 0= x and, (if b is not 1, then b= s(c) for some > some c) x+b= s(x+c) when b is not 0. æWe could > define multiplication of two such things by x*0= 0, > and x*b= x+ x*c for b not 0. æ æFor that particular system, with those > operations, yes, the empty set IS the number 0. æBut > there are many other ways to define numbers that do > not use sets as numbers. æThe important point is that > they are all equivalent- they all give the same > results. æYou can think of number in terms of any > one of them. But i claim i can calculate anything without > using zero as a number, OK. What's the value of sin(x) at x=pi? If I remove all the money from my bank account, > how much is in the account? > > Well then your out of money, you could say your > account is absent of > money but that hardly make 0 a number. Now it is starting to make sense. The reason for asking this question, and the reason you not responded to the repeated question what do you mean by 'a number' is that you have no idea what a number is! > > Please calculate these things without using > zero. æ æ æ æ - Randhy > === Subject: Re: Is the empty set a number? <30015947.1207832160209.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > On Apr 7, 12:24 pm, jonas.thornv...@hotmail.com On 7 Apr, 18:08, G.E. Ivey Do you understand the concept of equivalence? > æone way of defining the natural numbers is to set > 0 to be the empty set, {}, 1 to be the set whose > only member is the empty set, {{}}, 2 to be the set > whose only members are the empty set and {{}}- that > is, whose only members are 0 and 1- {0, 1}, etc. æ æWe can then define the successor of any > number, x, to be the set containing x and all of its > members and show that Peano's axioms for the natural > numbers hold. > We could define addition of two such things by > x+ 0= x and, (if b is not 1, then b= s(c) for some > some c) x+b= s(x+c) when b is not 0. æWe could > define multiplication of two such things by x*0= 0, > and x*b= x+ x*c for b not 0. æ æFor that particular system, with those > operations, yes, the empty set IS the number 0. æBut > there are many other ways to define numbers that do > not use sets as numbers. æThe important point is that > they are all equivalent- they all give the same > results. æYou can think of number in terms of any > one of them. But i claim i can calculate anything without > using zero as a number, OK. What's the value of sin(x) at x=pi? If I remove all the money from my bank account, > how much is in the account? Well then your out of money, you could say your > account is absent of > money but that hardly make 0 a number. æ Now it is starting to make sense. æThe reason for asking this question, and the reason you not responded to the repeated question what do you mean by 'a number' is that you have no idea what a number is! I can tell you what it not is, it is not a slice of of nothing, it is not a cut out of nothing, it is not a union of nothing, not a disjoint of nothing, not a part of a line with length 0, i could probably go on with this until tomorrow but i think i made the picture. Please calculate these things without using > zero. æ æ æ æ - Randhy- D.9alj citerad text - - Visa citerad text -- D.9alj citerad text - - Visa citerad text - === Subject: Re: Is the empty set a number? <30015947.1207832160209.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > On Apr 7, 12:24 pm, jonas.thornv...@hotmail.com On 7 Apr, 18:08, G.E. Ivey Do you understand the concept of equivalence? > æone way of defining the natural numbers is to set > 0 to be the empty set, {}, 1 to be the set whose > only member is the empty set, {{}}, 2 to be the set > whose only members are the empty set and {{}}- that > is, whose only members are 0 and 1- {0, 1}, etc. æ æWe can then define the successor of any > number, x, to be the set containing x and all of its > members and show that Peano's axioms for the natural > numbers hold. > We could define addition of two such things by > x+ 0= x and, (if b is not 1, then b= s(c) for some > some c) x+b= s(x+c) when b is not 0. æWe could > define multiplication of two such things by x*0= 0, > and x*b= x+ x*c for b not 0. æ æFor that particular system, with those > operations, yes, the empty set IS the number 0. æBut > there are many other ways to define numbers that do > not use sets as numbers. æThe important point is that > they are all equivalent- they all give the same > results. æYou can think of number in terms of any > one of them. But i claim i can calculate anything without > using zero as a number, OK. What's the value of sin(x) at x=pi? If I remove all the money from my bank account, > how much is in the account? Well then your out of money, you could say your > account is absent of > money but that hardly make 0 a number. æ Now it is starting to make sense. æThe reason for asking this question, and the reason you not responded to the repeated question what do you mean by 'a number' is that you have no idea what a number is! Please calculate these things without using > zero. æ æ æ æ - Randhy- D.9alj citerad text - - Visa citerad text -- D.9alj citerad text - - Visa citerad text - I have a working system to calculate, of course it have numbers zero just isn't in there and to be honest zero never was a number to start with human logic invented it. Although unnecessary it been around for along time. === Subject: Re: Is the empty set a number? <47fa8e69$0$22095$afc38c87@news.optusnet.com.au> posting-account=Xlhb0AoAAAAOPzB-kc6j7xEVDMNCmd4N Gecko/20080311 Firefox/2.0.0.13,gzip(gfe),gzip(gfe) Well i want you to prove that 0 is a number OK. Let's start with your definition of number > and your definition of 0. What do you mean > by 0? What do you mean by proving that > something is a number? - Randy I do not mean anything by 0 i have been taught that 0 is a number of > no value > A number is something that is necessary to represent a value > For me is absense of a value not a number, if x=0 4x seems to be a > nonsensical expression. ******************** > In answer to the question in your subject line, the empty set is not a > number. The empty set is a set, and whilst you haven't defined number I > would assume that you do not consider it as a set - this seems incompatible > with your definition of a number as a value. What logicians have done is to show that the structure which commences with > {} and creates other sets using the rule S(x) = x U {x} is isomorphic to the > definition of numbers given by Peano where S(y) = y+1. Then, by proving results about sets, we can prove results about numbers. Note that this definition of S(x) = x U {x} is arbitrary; other mappings are > possible, > including S(x) = {x}. Note that this isomorphism also does not require {} <-> 0. The same thing would work if we defined the isomorphism as {} <-> 1 (with the disadvantage we have no direct way of expressing zero) or {{}} <-> 0 (with the disadvantage that we have to type additional brace > characters all the time) So for reasons of simplicity, the most commonly used isomorphism is based > upon {} <-> 0 So no, {} is not a number. It can be associated with a number (as defined by > Peano) through a straightforward isomorphism, but equally other sets could > be mapped to zero, or {} could be mapped to some other number, and the > isomorphism would still hold. Its a logical but arbitrary choice that we > usually create the isomorphism using {} to represent 0. Hope this helps Peter Webb Why do you say S(x) = x U {x} is arbitrary? Isn't it necessary to build up the set? Do I misunderstand your notation? === Subject: Re: Is the empty set a number? posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q > Is the empty set a number? Historical, numbers are counts of objects/unities. E.g. an engine consists of 4 cylinders, 325 screws, and so on. An engine consists of zero apples. In a sense the zero stands for the absence of something and it defines a restclass of objects, namely the objects which are not part of a totality which is considered. In this elementary view there is always a number and an unity and we can say there is a multiplicative relation between number and unity: 4 times of cylinders 325 times of screws 0 times of apples Now 0 * something is nothing. So, in the case of zero the unity vanishes. The zero indicates absence, nothing. In difference to that, in set theory there is always something, at least the empty set. In this sense the numbers are more elementar than the sets, I think. Albrecht === Subject: Re: Is the empty set a number? > Is the empty set a number? > > Historical, numbers are counts of objects/unities. > E.g. an engine > consists of 4 cylinders, 325 screws, and so on. An > engine consists of > zero apples. > In a sense the zero stands for the absence of > something and it defines > a restclass of objects, namely the objects which are > not part of a > totality which is considered. > > In this elementary view there is always a number and > an unity and we > can say there is a multiplicative relation between > number and unity: > 4 times of cylinders > 325 times of screws > 0 times of apples > > Now 0 * something is nothing. So, in the case of zero > the unity > vanishes. The zero indicates absence, nothing. > In difference to that, in set theory there is always > something, at > least the empty set. > In this sense the numbers are more elementar than the > sets, I think. Your statement about numbers being more elementar than sets might in fact be trivial (although interesting) _within_ mathematics. As far as closed systems go, it could be proper to call _number_ the class of _tractability_. Set-based languages can be part of this recursive class, as indeed they seem to have shown since their very foundation. With an analogy, there we have the empty set peeked up as the starting point to transverse what, ultimately, is an identity class. It might be worth noting that each language (or representation), has got (does it?), in that strong constructive sense, its own kind of natural zero. Then, it seems we also have a legitimate definition of _zero_ as our _general_ starting point, as well as a general equivalence of closed languages. A stronger approach to the question of method (a sharpening of purpose, yet probably its restriction) might come from _outside_ mathematics (so to say, around infinity, since we have to keep using its language in the context of this discussion so, implicitly, of the sci.math domain). Here I show a _regression_ to which there is some logical/mathematical support, in terms of containment: measuring <= ordering <= distinguishing. That story on the child learning where right or left is, I find of great reference. It can help show how a _progression_ instead looks like; I mean, when seen from around the edge at infinity. Around that edge between mathematics and its implied foundation, the general theory of languages, all we have to do is preserve containment, by sticking within the bounds of our own very rules. That is, we cannot *directly* (analogically vs. digi-logically) _indicate_ absence until within the domain of hard (digi-logical) sciences. > The zero indicates absence, nothing. It might now be apparent how I would say that this sentence is incongruent and, strictly speaking, incorrect. Julio === Subject: Re: Is the empty set a number? Sorry, to clarify a bit: > > > That is, we cannot *directly* (analogically vs. > digi-logically) _indicate_ absence until within the > domain of hard (digi-logical) sciences. I meant to cover both analogical vs. digital and analogical vs. strictly logical. I guess I might have better ended up with something like metalogical... > > The zero indicates absence, nothing. > > It might now be apparent how I would say that this > sentence is incongruent and, strictly speaking, > incorrect. There cannot be a direct (non metalogical) construction for nothing of _anything_. Julio === Subject: Re: Is the empty set a number? > Sorry, to clarify a bit: > > > > > > That is, we cannot *directly* (analogically vs. > digi-logically) _indicate_ absence until within > the > domain of hard (digi-logical) sciences. > > I meant to cover both analogical vs. digital and > analogical vs. strictly logical. I guess I might have > better ended up with something like metalogical... > > > The zero indicates absence, nothing. > > It might now be apparent how I would say that this > sentence is incongruent and, strictly speaking, > incorrect. > > There cannot be a direct (non metalogical) > construction for nothing of _anything_. Indeed, we could write _it_ absence, zero, or empty, and even silence. In any case, the very fact we write (or say) _it_, makes us write (or say) the _metalogical_ Zero. Julio === Subject: Re: Is the empty set a number? <4515299.1207787821137.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Sorry, to clarify a bit: That is, we cannot *directly* (analogically vs. > digi-logically) indicate absence until within > the > domain of hard (digi-logical) sciences. I meant to cover both analogical vs. digital and > analogical vs. strictly logical. I guess I might have > better ended up with something like metalogical... The zero indicates absence, nothing. It might now be apparent how I would say that this > sentence is incongruent and, strictly speaking, > incorrect. There cannot be a direct (non metalogical) > construction for nothing of anything . Indeed, we could write it absence, zero, or empty, and even silence. In any case, the very fact we write (or say) it , makes us write (or say) the metalogical Zero. That you call it something or anything does not make it a mathematical entity, you have to prove you need it. Julio- D.9alj citerad text - - Visa citerad text - === Subject: Re: Is the empty set a number? Jonas: That you call it something or anything does not make it a mathematical entity, you have to prove you need it. *** That you name it and use it, IS exactly what makes a mathematical entity. Some terms in any axiomatic system are left undefined. Zero, number and successor are undefined terms in the Peano axioms. It would in fact be impossible to do mathematics without undefined terms. Do you know why? Tom === Subject: Re: Is the empty set a number? <29881705.1207815312020.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Jonas: That you call it something or anything does not make it a mathematical > entity, you have to prove you need it. *** That you name it and use it, IS exactly what makes > a mathematical entity. æSome terms in any axiomatic > system are left undefined. æZero, number and > successor are undefined terms in the Peano axioms. 4+0=4 isn't exactly using because if it was THE RESULT would be anything but 4. So no there is not a number 0 USED in the above operation. > It would in fact be impossible to do mathematics without > undefined terms. æDo you know why? It works just fine. > Tom === Subject: Re: Is the empty set a number? <29881705.1207815312020.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Jonas: That you call it something or anything does not make it a mathematical > entity, you have to prove you need it. *** That you name it and use it, IS exactly what makes > a mathematical entity. æSome terms in any axiomatic > system are left undefined. æZero, number and > successor are undefined terms in the Peano axioms. 4+0=4 isn't exactly using because if it was THE RESULT would be > anything but 4. So no there is not a number 0 USED in the above > operation. It would in fact be impossible to do mathematics without > undefined terms. æDo you know why? It works just fine. Tom- D.9alj citerad text - - Visa citerad text - It works just fine without zero === Subject: Re: Is the empty set a number? <4515299.1207787821137.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Sorry, to clarify a bit: That is, we cannot *directly* (analogically vs. > digi-logically) indicate absence until within > the > domain of hard (digi-logical) sciences. I meant to cover both analogical vs. digital and > analogical vs. strictly logical. I guess I might have > better ended up with something like metalogical... The zero indicates absence, nothing. It might now be apparent how I would say that this > sentence is incongruent and, strictly speaking, > incorrect. There cannot be a direct (non metalogical) > construction for nothing of anything . Indeed, we could write it absence, zero, or empty, and even silence. In any case, the very fact we write (or say) it , makes us write (or say) the metalogical Zero. > Julio- D.9alj citerad text - - Visa citerad text - Many layman people like me, actually beleive that there is zeroes in a number like 10,100,1000,10 000 when there actually is none, the zeroes are only misnomers inherited by the base and the imagined number of zero. If we had only one hand some stupid man probably come up with the 0,1,2,3,4,10,11,12,13,14,20,21,22,23,24 and so on, so of course the zeroes only is inherited property sign. It is a bit retarded to begin wiht to call anything something +0. It should of course be something +1. And it really can't be that hard to see....... === Subject: Re: Is the empty set a number? <4515299.1207787821137.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Sorry, to clarify a bit: That is, we cannot *directly* (analogically vs. > digi-logically) indicate absence until within > the > domain of hard (digi-logical) sciences. I meant to cover both analogical vs. digital and > analogical vs. strictly logical. I guess I might have > better ended up with something like metalogical... The zero indicates absence, nothing. It might now be apparent how I would say that this > sentence is incongruent and, strictly speaking, > incorrect. There cannot be a direct (non metalogical) > construction for nothing of anything . Indeed, we could write it absence, zero, or empty, and even silence. In any case, the very fact we write (or say) it , makes us write (or say) the metalogical Zero. > Julio- D.9alj citerad text - - Visa citerad text - Many layman people like me, actually beleive that there is zeroes in a > number like 10,100,1000,10 000 > when there actually is none, the zeroes are only misnomers inherited > by the base and the imagined number of zero. If we had only one hand > some stupid man probably come up with the > 0,1,2,3,4,10,11,12,13,14,20,21,22,23,24 and so on, so of course the > zeroes only is inherited property sign. It is a bit retarded to begin > wiht to call anything something +0. It should of course be something +1. And it really can't be that hard to see.......- D.9alj citerad text - - Visa citerad text - Ooops seems i lack a thumb :D === Subject: Re: Is the empty set a number? > On Apr 7, 11:46 am, jonas.thornv...@hotmail.com On 7 Apr, 17:36, jonas.thornv...@hotmail.com Is the empty set a number? I am also wondering if the empty set isn't a > missnaming? > For all i know {3}-{3}={} or it could be {3}-{3}= All this leaves me with one question what is the > set when there is no > values, and i must say it leaves me abit > cofuseed, is there really an > empty set? If so what is actually the set, i always thought > the group initself > was the set. > And there is no member in the group, there is no > group and no set, so > what is actually the empty set. JT Why are you starting so many different threads > on the same topic? Do you actually want to > discuss this topic or do you just want to > start a lot of threads and not participate in > any of them? - Randy- D.9alj citerad text - - Visa citerad text - > > Well i want you to prove that 0 is a number ????? One can't prove that any number is a number. Number is defined axiomatically. Look up Peano axioms or Dedekind-Peano and read up on the subject. Tom === Subject: Re: Is the empty set a number? <47fa8e69$0$22095$afc38c87@news.optusnet.com.au> posting-account=lHNboAoAAACyasQ0uqX7OeM_tLuWGoQp CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) - Randy I do not mean anything by 0 i have been taught that 0 is a number of > no value > A number is something that is necessary to represent a value Oh? I suggest you look at the p-adics for example. While they have a valuation, it will not correspond to anything that you or the OP would consider a value. === Subject: Re: Is the empty set a number? > Is the empty set a number? In the context of closed systems (and as far as I can tell): yes, quite trivially, because a set indeed is a number. Namely, whichever the symbolic system (or model) of your choice, what you are ultimately manipulating is but number objects, and the empty set is no exception, actually a natural foundational element for such systems. Julio === Subject: Re: Is the empty set a number? > On Apr 8, 1:49 pm, Julio Di Egidio > > [...] > > Sure. The psychological notion of quantity > is what one is trying to model---at least, > initially. Just as the psychological notion > of space is what one tries to model in > geometry. > What about the notion of a subject that -say- TOA > embeds? Could that be (already) enough to model > memory? I am, by the way, questioning your notion > of psychological notion, as applied here. > Incidentally, the notion of a subject could have > some bearings to the basic paradoxes in physics. > > I do not know what TOA is... Yet another three > leter acronym! > > Also, I do not know what you mean by > ``questioning your notion of psychological > notion, as applied here''. What I am saying > is that (all?) humans have an intuitive > notion of quantity and number, and that > the mathematical theory of numbers is > a model of that. This `psychological > notion' is probably what a platonist > would describe as the shadows in the > cave wall---but I am not a platonist ;-) Sorry, by questioning I mean something like not really criticizing yet not really asking for clarifications. To be straight, my idea here is: with psychological notion you indeed go off-domain. I have yet no idea about the role of psychology in a broader picture, but that it would probably belong to the unified field of social sciences. More foundational I suppose is philosophy, that is a closed theory on the ontology of reasoning. Anyway, if we stay within the domain of hard sciences, psychology has no role and all we need is a closed theory on the epistemology of reasoning. Of course, I will not deny there are implicit assumptions there, mainly on the roles of induction and purpose. > I do not think the connection with physics > will do anything to clarify the ontological > status of numbers: it has not done anything > to clarify the ontological status of pretty > `force' and so on share pretty much the > same status with `number'. Now it might be clear in which sense, in my opinion, it's not ontology at stake here. Physics seems to belong to the broader domain of hard sciences, so ultimately concerned with an epistemological foundation. It's mathematics that becomes an ambiguous term here. Julio === Subject: Re: Is the empty set a number? > On Apr 8, 9:00æam, Julio Di Egidio > Aatu Koskensilta (aatu.koskensi...@xortec.fi) Wovon man nicht sprechen kann, daruber muss > man > schweigen > æ- Ludwig Wittgenstein, Tractatus > Logico-Philosophicus Your Wittgegstein quote seems quite relevant to > the > question, What is > a number?. I agree it is relevant, I do not agree it closes > the argument. (Ludwig didn't really finish there.) Julio > > I'm not worried about is actually a number. The > notion of number will > fleshed out only by using various representations of > numbers and > understanding the implications of our models of > numbers. Yep, couldn't agree more! I'd stress: implications... Julio === Subject: Re: Is the empty set a number? > On Apr 8, 12:39 pm, Aatu Koskensilta What *is* a number? What is it that is prior to any representation? What the naturals are, for example, is explained by saying that they > are what one obtains from 0 by repeatedly applying the 'add > one'-operation, i.e. 0, 1, 2, ... and so on. This, of course, is not > a definition; rather, it's an explanation we all must understand > before we can understand anything in mathematics. It is also not an > answer to the question What is a number? in any philosophically > interesting sense. However, this basic mathematical understanding is > prior to any representation -- indeed, if we didn't have that > understanding we'd have no criteria by which to judge the adequacy of > any representation. > > Sure. The psychological notion of quantity > is what one is trying to model---at least, > initially. Just as the psychological notion > of space is what one tries to model in > geometry. > > But as soon as one does that, one of course notices that the choice > of specific representation was arbitrary, making it plain that it is > pointless. Pointless as an answer to the question What is a number?, yes, but > not at all pointless from a mathematical perspective. > > Of course not, because the fact that > one can pick a specific construction > proves in particular that one can pick > *a* construction, thereby proving > non-vacuity of the theory (relative to > the consistency of whatever context one > is using to do the construction, of > course...) > > Also, the construction of such concrete > models allows one to build intuitions. > > But I would say that for all other > purposes, the Dedekind construction of > the reals, say, is absolutely irrelevant > from a mathematical perspective. This paragraph I do not understand. I think that the Dedekind construction of the real numbers is entertained exactly for its relevance to mathematics. -- Michael Press === Subject: Re: Is the empty set a number? > On Apr 8, 12:39 pm, Aatu Koskensilta .................. > Of course not, because the fact that > one can pick a specific construction > proves in particular that one can pick > *a* construction, thereby proving > non-vacuity of the theory (relative to > the consistency of whatever context one > is using to do the construction, of > course...) > Also, the construction of such concrete > models allows one to build intuitions. > But I would say that for all other > purposes, the Dedekind construction of > the reals, say, is absolutely irrelevant > from a mathematical perspective. >This paragraph I do not understand. I think that the >Dedekind construction of the real numbers is entertained >exactly for its relevance to mathematics. The Dedekind construction of the reals is ONE way of constructing the reals, and definitely not the only way. Just as for the integers, which representation is used is not relevant. One could instead use expansions of the fractional part to an arbitrary base; there is less of a problem in defining addition and multiplication, although this is not too great a problem in using Dedekind cuts for the positive reals. Also, one could use continued fractions, and I would not be surprised if a dozen other methods have not already been used. Cauchy sequences are another method, although the justification of their existence is needed. The expansion in a given base is one way to justify it. Cauchy sequences are a much more intuitive way to consider reals, and I believe this is the first one presented. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Is the empty set a number? posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU 5.1),gzip(gfe),gzip(gfe) > Can you tell me if two empty sets equals one empty set i would be > greatful for an answer. By the Axiom of Extensionality in ZFC, all empty sets are equal. It appears that the OP has one of two concerns: 1. Is zero really a number? 2. Is the empty set really a set? And these two questions have distinct answers. 1. Many of the others have already discussed to which objects we assign the property of numberhood in this thread. But the bottom line is that any ring must have an additive identity, and that element is zero. Indeed, every semigroup can be made into a monoid by simply attaching such a neutral element. To standard mathematicians, the unqualified word number refers to an element of C, the set of complex numbers. Thus zero is a number since 0eC. One might ask what the set of all numbers is to the OP. If we let J (for Jonas) be the set of all numbers that the OP accepts, then clearly J is a proper subset of C, since 0eC but yet ~0eJ. So what numbers are in J? I doubt that the OP accepts the existence of negative numbers, since then one would have to wonder what -1+1 is. Historically the Greeks and Romans, just like the OP, denied the existence of zero, but the Sumerians, Mayans, and Hindus all had symbols for zero. The largest set one can have without the existence of zero is the set of unsigned reals R+, which is labeled by a script P in Metamath. Notice how Metamath develops the unsigned fractions and unsigned reals via Dedekind cuts well before developing zero and signed numbers. This matches the historical development, where Pythagoras discovered sqrt(2) over 2000 years before Cardano introduced negative numbers. Kronecker said that God created the integers -- but it's uncertain whether he meant the positive natural numbers or the signed integers. But of course, one trick is simply to let a signed integer simply be an equivalence class of ordered pairs of natural numbers, as is usually done, so that 0 = {(1,1),(2,2),(3,3),...}. 2. But if you deny the existence of the empty set, then you have a deeper problem than if you merely deny the numberhood of zero. For the Axiom of Foundation (AKA Regularity) states that every set is based on the empty set, in that every set has the empty set as an element of its transitive closure. So if you don't want an empty set then you must deny Foundation/Regularity. One sci.logic poster, Zuhair, also wanted to come up with a set theory once in which there is no empty set. But unfortunately, he was not able to come up with such a theory in a way that it would be consistent. === Subject: Re: Is the empty set a number? posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Can you tell me if two empty sets equals one empty set i would be > greatful for an answer. By the Axiom of Extensionality in ZFC, all empty sets are equal. It appears that the OP has one of two concerns: 1. Is zero really a number? > 2. Is the empty set really a set? And these two questions have distinct answers. 1. Many of the others have already discussed to which objects > we assign the property of numberhood in this thread. But the > bottom line is that any ring must have an additive identity, > and that element is zero. Indeed, every semigroup can be made > into a monoid by simply attaching such a neutral element. To standard mathematicians, the unqualified word number > refers to an element of C, the set of complex numbers. Thus > zero is a number since 0eC. One might ask what the set of all numbers is to the OP. If > we let J (for Jonas) be the set of all numbers that the OP > accepts, then clearly J is a proper subset of C, since 0eC > but yet ~0eJ. So what numbers are in J? I doubt that the OP accepts the > existence of negative numbers, since then one would have > to wonder what -1+1 is. Well that would be dent to no accept negative numbers of course i do. What i say is that the result from the transaction 1-1 is not a number, and that is quite another issue. If you accept this and build your numbersystem and architecture supporting this. You do not have to think about division by zero. And it still will give perfectly valid result for any calculation. >Historically the Greeks and Romans, > just like the OP, denied the existence of zero, but the > Sumerians, Mayans, and Hindus all had symbols for zero. Yes i am aware of that, and i think it .92s a problem inherited by our logic to want to put number to nothing. Because nothing really do not have a *value*. Representing a number as base dependent postional value sets works very good for any base. Here is some expamples in DECIMAL base using positional value representation and as you can guess 0 is missing representation in this system. -2000500009=-{[A,2][5,5][1,9]} 9000020005={[A,9][5,2][1.6]} 0,50000000009={[-1,5][-B,9]} -1.90005=-{[1,1][-1,9][-5,5]} > The largest set one can have without the existence of zero > is the set of unsigned reals R+, which is labeled by a > script P in Metamath. Notice how Metamath develops the > unsigned fractions and unsigned reals via Dedekind cuts > well before developing zero and signed numbers. This > matches the historical development, where Pythagoras > discovered sqrt(2) over 2000 years before Cardano > introduced negative numbers. Kronecker said that God created the integers -- but it's > uncertain whether he meant the positive natural numbers or > the signed integers. But of course, one trick is simply > to let a signed integer simply be an equivalence class of > ordered pairs of natural numbers, as is usually done, so > that 0 = {(1,1),(2,2),(3,3),...}. 2. But if you deny the existence of the empty set, then > you have a deeper problem than if you merely deny the > numberhood of zero. For the Axiom of Foundation (AKA > Regularity) states that every set is based on the empty > set, in that every set has the empty set as an element of > its transitive closure. So if you don't want an empty set > then you must deny Foundation/Regularity. One sci.logic poster, Zuhair, also wanted to come up with > a set theory once in which there is no empty set. But > unfortunately, he was not able to come up with such a > theory in a way that it would be consistent. You try this === Subject: Re: Is the empty set a number? posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > So what numbers are in J? I doubt that the OP accepts the > existence of negative numbers, since then one would have > to wonder what -1+1 is. > Well that would be dent to no accept negative numbers of course i do. That's interesting, since, as I said earlier, most opponents of zero rejected the negative numbers as well. > Here is some expamples in DECIMAL base using positional value > representation and as you can guess 0 is missing representation in > this system. The OP may be interested that Wikipedia gives another decimal notation that also avoids the use of zero: http://en.wikipedia.org/wiki/Bijective numeration In the bijective base ten system, the numbers one through nine are the same as in standard decimal, but the number ten must differ since we want to avoid a zero. So we need a new symbol. Since the OP uses the letter A to represent ten in his numeration system, we shall do this same in this bijective system. The numbers 11 through 19 are the same as in the standard system, then 1A is used for twenty. Further examples given by Wikipedia are: [C]onventional 100 becomes 9A, conventional 101 becomes A1, conventional 302 becomes 2A2, conventional 1000 becomes 99A, conventional 1110 becomes AAA, conventional 2010 becomes 19AA, and so on. And here are the OP's examples written in bijective decimal: > -2000500009=-{[A,2][5,5][1,9]} -2000500009 = -199A4999A9 > 9000020005={[A,9][5,2][1.6]} 9000020005 = 8999A199A5 Notice that, to mention another sci.math poster, Ross Finlayson's unary numeration is actually bijective base-one, and is also mentioned in the Wikipedia link. The major drawback with bijective numeration systems is that non-integral numbers are a bit awkward to work with. Suppose, for example, that we extend bijective numeration in the natural way to include non-integers. Since, according to Wikipedia: 302 = 2A2 we would naturally conclude: 3.02 = 2.A2 and therefore 2.A2 > 3, a non-obvious inequality. Similarly: > 0,50000000009={[-1,5][-B,9]} 0.50000000009 = .499999999A9 > .5 > -1.90005=-{[1,1][-1,9][-5,5]} -1.90005 = -1.899A5 < -1.9 and so on. And what's worse is that, in general, positive decimals less than 1/9 can't be represented -- with a few exceptions such as .1, .11, .111, etc. Certainly no positive decimal less than 1/A can be represented. The problem, as we can see, is that in order to compare two decimals, one must make them the same length. We can do this in the standard system simply by adding zeros to the number with fewer decimal places, but in the bijective system, there is no zero. One actually has to change some of the digits of a number in order to make them the same length, so when comparing 2.A2 to 3, the latter must become 2.9A, so that it is now clearly less than 2.A2. Indeed, all decimals must be written with infinitely many digits in order to compare them in general. The number 1, for example, must be written as the infinite decimal .999..., at which point it is obviously less than .999A999..., but greater than .9998A999.... Notice how 1.000... = .999... controversy that has dominated many sci.math threads is avoided -- as it turns out, the number one has two infinite decimal expansions in the standard system, but only one such expansion, namely .999..., in the bijective system. Of course, 1 has infinitely many terminating expansions in both the standard and the bijective systems, such as 1.0, 1.00, 1.000 in the former and .A, .9A, .99A in the latter. It's the numbers such as A/9 which have two expansions in the bijective system -- .AAA... and 1.111.... At the end of the Wikipedia page is a link to RR Forslund, who proposes replace the existing numeration system with an alternative numeration system (bijective). What does Forslund say about the non-integer problem? He appears to be opposed to decimal expansions for numbers that are not integers altogether. (This is in stark contrast to a thread here at sci.math about two months ago, which was titled, Abolish fractions? Forslund would abolish decimals!) === Subject: Re: Is the empty set a number? posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Can you tell me if two empty sets equals one empty set i would be > greatful for an answer. By the Axiom of Extensionality in ZFC, all empty sets are equal. It appears that the OP has one of two concerns: 1. Is zero really a number? > 2. Is the empty set really a set? And these two questions have distinct answers. 1. Many of the others have already discussed to which objects > we assign the property of numberhood in this thread. But the > bottom line is that any ring must have an additive identity, > and that element is zero. Indeed, every semigroup can be made > into a monoid by simply attaching such a neutral element. To standard mathematicians, the unqualified word number > refers to an element of C, the set of complex numbers. Thus > zero is a number since 0eC. One might ask what the set of all numbers is to the OP. If > we let J (for Jonas) be the set of all numbers that the OP > accepts, then clearly J is a proper subset of C, since 0eC > but yet ~0eJ. So what numbers are in J? I doubt that the OP accepts the > existence of negative numbers, since then one would have > to wonder what -1+1 is. Well that would be dent to no accept negative numbers of course i do. > What i say is that the result from the transaction 1-1 is not a > number, and that is quite another issue. If you accept this and build > your numbersystem and architecture supporting this. You do not have to > think about division by zero. And it still will give perfectly valid > result for any calculation. Historically the Greeks and Romans, > just like the OP, denied the existence of zero, but the > Sumerians, Mayans, and Hindus all had symbols for zero. Yes i am aware of that, and i think it .92s a problem inherited by our > logic to want to put number to nothing. Because nothing really do not > have a *value*. Representing a number as base dependent postional value sets works > very good for any base. Here is some expamples in DECIMAL base using positional value > representation and as you can guess 0 is missing representation in > this system. -2000500009=-{[A,2][5,5][1,9]} > 9000020005={[A,9][5,2][1.6]} 0,50000000009=ooops again{[-1,5][-B,9]}should of course be A1,9 > -1.90005=-{[1,1][-1,9][-5,5]} The largest set one can have without the existence of zero > is the set of unsigned reals R+, which is labeled by a > script P in Metamath. Notice how Metamath develops the > unsigned fractions and unsigned reals via Dedekind cuts > well before developing zero and signed numbers. This > matches the historical development, where Pythagoras > discovered sqrt(2) over 2000 years before Cardano > introduced negative numbers. Kronecker said that God created the integers -- but it's > uncertain whether he meant the positive natural numbers or > the signed integers. But of course, one trick is simply > to let a signed integer simply be an equivalence class of > ordered pairs of natural numbers, as is usually done, so > that 0 = {(1,1),(2,2),(3,3),...}. 2. But if you deny the existence of the empty set, then > you have a deeper problem than if you merely deny the > numberhood of zero. For the Axiom of Foundation (AKA > Regularity) states that every set is based on the empty > set, in that every set has the empty set as an element of > its transitive closure. So if you don't want an empty set > then you must deny Foundation/Regularity. One sci.logic poster, Zuhair, also wanted to come up with > a set theory once in which there is no empty set. But > unfortunately, he was not able to come up with such a > theory in a way that it would be consistent. You try this- D.9alj citerad text - - Visa citerad text -- D.9alj citerad text - - Visa citerad text - === Subject: Re: Is the empty set a number? posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Can you tell me if two empty sets equals one empty set i would be > greatful for an answer. By the Axiom of Extensionality in ZFC, all empty sets are equal. It appears that the OP has one of two concerns: 1. Is zero really a number? > 2. Is the empty set really a set? And these two questions have distinct answers. 1. Many of the others have already discussed to which objects > we assign the property of numberhood in this thread. But the > bottom line is that any ring must have an additive identity, > and that element is zero. Indeed, every semigroup can be made > into a monoid by simply attaching such a neutral element. To standard mathematicians, the unqualified word number > refers to an element of C, the set of complex numbers. Thus > zero is a number since 0eC. One might ask what the set of all numbers is to the OP. If > we let J (for Jonas) be the set of all numbers that the OP > accepts, then clearly J is a proper subset of C, since 0eC > but yet ~0eJ. So what numbers are in J? I doubt that the OP accepts the > existence of negative numbers, since then one would have > to wonder what -1+1 is. Well that would be dent to no accept negative numbers of course i do. > What i say is that the result from the transaction 1-1 is not a > number, and that is quite another issue. If you accept this and build > your numbersystem and architecture supporting this. You do not have to > think about division by zero. And it still will give perfectly valid > result for any calculation. Historically the Greeks and Romans, > just like the OP, denied the existence of zero, but the > Sumerians, Mayans, and Hindus all had symbols for zero. Yes i am aware of that, and i think it .92s a problem inherited by our > logic to want to put number to nothing. Because nothing really do not > have a *value*. Representing a number as base dependent postional value sets works > very good for any base. Here is some expamples in DECIMAL base using positional value > representation and as you can guess 0 is missing representation in > this system. -2000500009=-{[A,2][5,5][1,9]} > 9000020005={[A,9][5,2][1.6]} æ0,50000000009=ooops again{[-1,5][-B,9]}should of course be A1,9 -1.90005=-{[1,1][-1,9][-5,5]} The largest set one can have without the existence of zero > is the set of unsigned reals R+, which is labeled by a > script P in Metamath. Notice how Metamath develops the > unsigned fractions and unsigned reals via Dedekind cuts > well before developing zero and signed numbers. This > matches the historical development, where Pythagoras > discovered sqrt(2) over 2000 years before Cardano > introduced negative numbers. Kronecker said that God created the integers -- but it's > uncertain whether he meant the positive natural numbers or > the signed integers. But of course, one trick is simply > to let a signed integer simply be an equivalence class of > ordered pairs of natural numbers, as is usually done, so > that 0 = {(1,1),(2,2),(3,3),...}. 2. But if you deny the existence of the empty set, then > you have a deeper problem than if you merely deny the > numberhood of zero. For the Axiom of Foundation (AKA > Regularity) states that every set is based on the empty > set, in that every set has the empty set as an element of > its transitive closure. So if you don't want an empty set > then you must deny Foundation/Regularity. One sci.logic poster, Zuhair, also wanted to come up with > a set theory once in which there is no empty set. But > unfortunately, he was not able to come up with such a > theory in a way that it would be consistent. You try this- D.9alj citerad text - - Visa citerad text -- D.9alj citerad text - - Visa citerad text -- D.9alj citerad text - - Visa citerad text -- D.9alj citerad text - - Visa citerad text - Piuh noone spotted that one should be 1A === Subject: Re: Is the empty set a number? posting-account=kxPkPAoAAACjJi8w0gL9bnyznPzdw9HW Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > Can you tell me if two empty sets equals one empty set i would be > greatful for an answer. By the Axiom of Extensionality in ZFC, all empty sets are equal. It appears that the OP has one of two concerns: 1. Is zero really a number? > 2. Is the empty set really a set? And these two questions have distinct answers. 1. Many of the others have already discussed to which objects > we assign the property of numberhood in this thread. But the > bottom line is that any ring must have an additive identity, > and that element is zero. Indeed, every semigroup can be made > into a monoid by simply attaching such a neutral element. To standard mathematicians, the unqualified word number > refers to an element of C, the set of complex numbers. Thus > zero is a number since 0eC. One might ask what the set of all numbers is to the OP. If > we let J (for Jonas) be the set of all numbers that the OP > accepts, then clearly J is a proper subset of C, since 0eC > but yet ~0eJ. So what numbers are in J? I doubt that the OP accepts the > existence of negative numbers, since then one would have > to wonder what -1+1 is. Well that would be dent to no accept negative numbers of course i do. > What i say is that the result from the transaction 1-1 is not a > number, and that is quite another issue. If you accept this and build > your numbersystem and architecture supporting this. You do not have to > think about division by zero. And it still will give perfectly valid > result for any calculation. Historically the Greeks and Romans, > just like the OP, denied the existence of zero, but the > Sumerians, Mayans, and Hindus all had symbols for zero. Yes i am aware of that, and i think it .92s a problem inherited by our > logic to want to put number to nothing. Because nothing really do not > have a *value*. Representing a number as base dependent postional value sets works > very good for any base. Here is some expamples in DECIMAL base using positional value > representation and as you can guess 0 is missing representation in > this system. -2000500009=-{[A,2][5,5][1,9]} > 9000020005={[A,9][5,2][1.6]} > 0,50000000009={[-1,5][-B,9]} -1.90005=-oooops-{[1,1][-1,9][-5,5]} The largest set one can have without the existence of zero > is the set of unsigned reals R+, which is labeled by a > script P in Metamath. Notice how Metamath develops the > unsigned fractions and unsigned reals via Dedekind cuts > well before developing zero and signed numbers. This > matches the historical development, where Pythagoras > discovered sqrt(2) over 2000 years before Cardano > introduced negative numbers. Kronecker said that God created the integers -- but it's > uncertain whether he meant the positive natural numbers or > the signed integers. But of course, one trick is simply > to let a signed integer simply be an equivalence class of > ordered pairs of natural numbers, as is usually done, so > that 0 = {(1,1),(2,2),(3,3),...}. 2. But if you deny the existence of the empty set, then > you have a deeper problem than if you merely deny the > numberhood of zero. For the Axiom of Foundation (AKA > Regularity) states that every set is based on the empty > set, in that every set has the empty set as an element of > its transitive closure. So if you don't want an empty set > then you must deny Foundation/Regularity. One sci.logic poster, Zuhair, also wanted to come up with > a set theory once in which there is no empty set. But > unfortunately, he was not able to come up with such a > theory in a way that it would be consistent. You try this- D.9alj citerad text - - Visa citerad text -- D.9alj citerad text - - Visa citerad text - === Subject: Re: Is the empty set a number? > On Apr 7, 2:26æpm, jonas.thornv...@hotmail.com > Can you tell me if two empty sets equals one empty > set i would be > greatful for an answer. > > By the Axiom of Extensionality in ZFC, all empty sets > are equal. > > It appears that the OP has one of two concerns: > > 1. Is zero really a number? > 2. Is the empty set really a set? IMHO, no. The original question is: > Is the empty set a number? With your splitting above, you seem to have lost the very nature of the question. Paralogism. Julio === Subject: Re: Is the empty set a number? <78294.1207763017973.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > It appears that the OP has one of two concerns: > 1. Is zero really a number? > 2. Is the empty set really a set? > IMHO, no. The original question is: > Is the empty set a number? I already know that, and I addressed this in the first part of the post that you snipped. To put it concisely, I propose that one should consider: The empty set is a number. This sentence is true, with the usual constructions of the various types of numbers, if one fills in the blank with the words cardinal or ordinal, but false if one fills it in with complex or real instead. But then again, in standard analysis, the unqualified word number means complex number unless otherwise specified. So where did I get questions 1 and 2 from, anyway? I inferred both of them from comments the OP made elsewhere in this thread as well as in other threads. In particular, the OP questioned the sethood of the empty set in the second post of this thread, while he questioned the numberhood of zero several times in this thread, as well as in another thread. === Subject: Re: Is the empty set a number? >On Apr 8, 12:10 pm, Aatu Koskensilta We cannot talk about what IS a number, but only what REPRESENTS a > number. > Numbers are prior to any representation. In particular, our > understanding of the naturals, and other such finitary inductively > defined stuff, is more fundamental than any set theoretic > representation. >What *is* a number? What is it that is prior >to any representation? >There is a psychological notion of number, >which may or may not be prior to its >representations, but that's a completely >different `number'. I remember, for example, >an rather interesting book by Piaget >on the subject. There are many concepts which apply to the counting numbers (integers), and it is a mistake to pick out one of them and call it THE concept of number. The new math picked the cardinal representation. I would place the ordinal definition first, and bring in the cardinal representation instead. One reason is that the ordinal representation is actually more simple, although a mathematician might not think so. Another is that the ordinal definition is self-contained, whereas even to define a finite set requires ordering. >Mathematically, I do not think one can >make *any* sense of the question `what >is a number', unless one is willing to >pick a specific representation (ie, to >fix a specific construction for >numbers) and stick with them as a definition. Once one shows that the models are isomorphic, this is no longer the case. This means that it a representation satisfies the conditions, all properties of the numbers hold in it. Other statements, such as 2 and 3 start with the same letter involves the representation. >But as soon as one does that, one >of course notices that the choice of >specific representation was arbitrary, >making it plain that it is pointless. >-- m -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Is the empty set a number? > <2308039c-e114-43d9-ab27-7d3f30605f98@y21g2000hsf.goog > legroups.com>, > =?ISO-8859-1?Q?Mariano_Su=E1rez=2DAlvarez?= >On Apr 8, 12:10 pm, Aatu Koskensilta > We cannot talk about what IS a number, but only > what REPRESENTS a > number. > > Numbers are prior to any representation. In > particular, our > understanding of the naturals, and other such > finitary inductively > defined stuff, is more fundamental than any set > theoretic > representation. > >What *is* a number? What is it that is prior >to any representation? > >There is a psychological notion of number, >which may or may not be prior to its >representations, but that's a completely >different `number'. I remember, for example, >an rather interesting book by Piaget >on the subject. > > There are many concepts which apply to the > counting numbers (integers), and it is a mistake > to pick out one of them and call it THE concept > of number. The new math picked the cardinal > representation. I would place the ordinal > definition first, and bring in the cardinal > representation instead. One reason is that the > ordinal representation is actually more simple, > although a mathematician might not think so. > Another is that the ordinal definition is > self-contained, whereas even to define a finite > set requires ordering. Couldn't your argument (in this case and in principle) be enough to give preminence to (in this case) the ordinal definition? In fact (and, although work in progress, to Your revision), up to here I seem to have a progression of the sort: i) distinction (inequalities (dis-comparability)); ii) ordinality (relation (subjectivity)); iii) cardinality (measure (objectivity)). > >Mathematically, I do not think one can >make *any* sense of the question `what >is a number', unless one is willing to >pick a specific representation (ie, to >fix a specific construction for >numbers) and stick with them as a definition. > > Once one shows that the models are isomorphic, > this is no longer the case. This means that > it a representation satisfies the conditions, > all properties of the numbers hold in it. > Other statements, such as 2 and 3 start with > the same letter involves the representation. Indeed, one thing is ask what a number is, other is ask < < <. Naively, I'd say: in the first instance we are asking from within mathematics (from within <), where an answer is the tautological closure of mathematics (basically uninteresting); in the second instance we are asking at a higher level: for our purpose, in the second instance we are asking from within the level of (hard) science (from within <<...???...>), where an answer is the self-referential foundation for mathematics (ultimately uninteresting). > >But as soon as one does that, one >of course notices that the choice of >specific representation was arbitrary, >making it plain that it is pointless. The choice of representation in a first instance may be utterly uninteresting. Yet, if we don't forget that representation mediates our actual usage of mathematics, again the picture is not so just closed. Julio === Subject: Re: Is the empty set a number? <6WLKj.323134$gR2.233579@reader1.news.saunalahti.fi> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) On Apr 8, 12:10 pm, Aatu Koskensilta We cannot talk about what IS a number, but only what REPRESENTS a > number. > Numbers are prior to any representation. In particular, our > understanding of the naturals, and other such finitary inductively > defined stuff, is more fundamental than any set theoretic > representation. >What *is* a number? What is it that is prior >to any representation? >There is a psychological notion of number, >which may or may not be prior to its >representations, but that's a completely >different `number'. I remember, for example, >an rather interesting book by Piaget >on the subject. There are many concepts which apply to the > counting numbers (integers), and it is a mistake > to pick out one of them and call it THE concept > of number. The new math picked the cardinal > representation. I would place the ordinal > definition first, and bring in the cardinal > representation instead. One reason is that the > ordinal representation is actually more simple, > although a mathematician might not think so. > Another is that the ordinal definition is > self-contained, whereas even to define a finite > set requires ordering. Mathematically, I do not think one can >make *any* sense of the question `what >is a number', unless one is willing to >pick a specific representation (ie, to >fix a specific construction for >numbers) and stick with them as a definition. Once one shows that the models are isomorphic, > this is no longer the case. This means that > it a representation satisfies the conditions, > all properties of the numbers hold in it. > Other statements, such as 2 and 3 start with > the same letter involves the representation. That was precisely my point. There is no point in choosing a particular representation, because all representations are isomorphic and nothing one wants to do will depend on the representation chosen. Hence, it really does not make any sense to ask `what is a number'. -- m === Subject: Re: Is the empty set a number? > That was precisely my point. There is no > point in choosing a particular representation, > because all representations are isomorphic > and nothing one wants to do will depend > on the representation chosen. Hence, it > really does not make any sense to ask > `what is a number'. So you think those who ask What is a number? are asking What representation for numbers should we choose?? -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Is the empty set a number? > That was precisely my point. There is no > point in choosing a particular representation, > because all representations are isomorphic > and nothing one wants to do will depend > on the representation chosen. Hence, it > really does not make any sense to ask > `what is a number'. >So you think those who ask What is a number? are asking What >representation for numbers should we choose?? One should be familiar with sufficiently many of the representations that one can be said to understand the integers. The cardinal and ordinal representations are the most basic ones; magnitude is an interpretation of the cardinal one, and induction is a key property of the ordinal one. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Is the empty set a number? > One should be familiar with sufficiently many of the > representations that one can be said to understand > the integers. Why? How do the set theoretic representations enter into our understanding of the integers? -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Is the empty set a number? <%p0Lj.323478$6w7.257179@reader1.news.saunalahti.fi> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) On Apr 10, 2:27 pm, Aatu Koskensilta One should be familiar with sufficiently many of the > representations that one can be said to understand > the integers. Why? How do the set theoretic representations enter into our > understanding of the integers? In the same way as the models of mathematical physics enter into our understanding of the physical world? -- m === Subject: Re: Is the empty set a number? > On 2008-04-08, in sci.math, Mariano Su.87rez-Alvarez > That was precisely my point. There is no > point in choosing a particular representation, > because all representations are isomorphic > and nothing one wants to do will depend > on the representation chosen. Hence, it > really does not make any sense to ask > `what is a number'. > > So you think those who ask What is a number? are > asking What > representation for numbers should we choose?? > Given that meaning is independent of language, the question is proper. Tom > -- > Aatu Koskensilta (aatu.koskensilta@xortec.fi) > > Wovon man nicht sprechen kann, daruber muss man > schweigen > - Ludwig Wittgenstein, Tractatus > s Logico-Philosophicus === Subject: Re: Is the empty set a number? <6WLKj.323134$gR2.233579@reader1.news.saunalahti.fi> <%p0Lj.323478$6w7.257179@reader1.news.saunalahti.fi> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/20080325 Fedora/2.0.0.13-1.fc8 Firefox/2.0.0.13,gzip(gfe),gzip(gfe) On Apr 9, 6:57 am, Aatu Koskensilta That was precisely my point. There is no > point in choosing a particular representation, > because all representations are isomorphic > and nothing one wants to do will depend > on the representation chosen. Hence, it > really does not make any sense to ask > `what is a number'. So you think those who ask What is a number? are asking What > representation for numbers should we choose?? Not at all. I think that those who ask What is a number? are asking something akin to Where does the rainbow end?. I am saying that it does not make any sense to say that an object is a number but to say that it is a member of a class of objects with certain operations and properties (say, a complete ordered field, or the prime field of characteristic zero, or a peano system, etc). Numbers do not exist individually: what exists is number *systems* or, more interesting, isomorphism classes of number systems. *Every* mathematical object X can be the square root of the result of adding the multiplicative unit to itself in a complete ordered field: take any complete ordered field F, and consider the set G = ( F x { X } - { ( sqrt(2), X ) } ) union { X } There is an obvious bijection F --> G, which one can trivially use to turn G into an ordered field. It will be, of course, be a complete ordered field. Therefore the object X with which we started is a real number is the trivial sense that it belongs to a complete ordered field. Clearly, this is an utterly uninteresting fact! -- m === Subject: Re: Is the empty set a number? <6WLKj.323134$gR2.233579@reader1.news.saunalahti.fi> posting-account=7Ew-gAoAAACK7aClJ3SDozB1-FCoCyqu .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.1),gzip(gfe),gzip(gfe) On Apr 8, 2:14æpm, Mariano Su.87rez-Alvarez >On Apr 8, 12:10 pm, Aatu Koskensilta We cannot talk about what IS a number, but only what REPRESENTS a > number. > Numbers are prior to any representation. In particular, our > understanding of the naturals, and other such finitary inductively > defined stuff, is more fundamental than any set theoretic > representation. >What *is* a number? What is it that is prior >to any representation? >There is a psychological notion of number, >which may or may not be prior to its >representations, but that's a completely >different `number'. I remember, for example, >an rather interesting book by Piaget >on the subject. There are many concepts which apply to the > counting numbers (integers), and it is a mistake > to pick out one of them and call it THE concept > of number. æThe new math picked the cardinal > representation. æI would place the ordinal > definition first, and bring in the cardinal > representation instead. æOne reason is that the > ordinal representation is actually more simple, > although a mathematician might not think so. > Another is that the ordinal definition is > self-contained, whereas even to define a finite > set requires ordering. Mathematically, I do not think one can >make *any* sense of the question `what >is a number', unless one is willing to >pick a specific representation (ie, to >fix a specific construction for >numbers) and stick with them as a definition. Once one shows that the models are isomorphic, > this is no longer the case. æThis means that > it a representation satisfies the conditions, > all properties of the numbers hold in it. > Other statements, such as 2 and 3 start with > the same letter involves the representation. That was precisely my point. There is no > point in choosing a particular representation, > because all representations are isomorphic > and nothing one wants to do will depend > on the representation chosen. Hence, it > really does not make any sense to ask > `what is a number'. -- m- Hide quoted text - - Show quoted text - That really is the point: all representations are isomorphic so at level there is no difference. However, when dealing with a particular problem one representation might preferable to another on the grounds of simplifying your work. === Subject: Re: Is the empty set a number? <23765014.1207678547714.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=7Ew-gAoAAACK7aClJ3SDozB1-FCoCyqu .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.1),gzip(gfe),gzip(gfe) > On Apr 8, 9:49æam, Julio Di Egidio > On Apr 8, 12:39 pm, Aatu Koskensilta > But I would say that for all other > purposes, the Dedekind construction of > the reals, say, is absolutely irrelevant > from a mathematical perspective. I would agree on that, as much as -it- is > completely embedded in mathematics itself. I'd say > asking what a number is, is itself asking about the > connection between real world and the world of > numbers, though asking it from within mathematics. > My... intuition, here, is that a step backwards, by > connecting physics to mathematics at the scientific > level, we could answer that question on the status of > the so called numbers in a consistent and natural > way. This reminds me of the time that geometers believed > that there was > only Euclidean geometry before counterexamples were > produced. > Progress in geometry really leaped ahead once the > apparent link to > real world geometry was severed and the apparent > obviousness of the > postulates were recognized to be somewhat arbitrary. Pardon me, but IMHO you risk here not to be thinking out of the box. We need give counter-examples to the closed systems concept, that I would agree (although there cannot really be counter-examples, by construction, which is a matter for another post). In fact, there is a distinction between Euclidean geometry as an (unclosed) mathematical system, subject to the progress inflicted by goedelization, and the Euclidean concept of Geometry, which is another way to tell that progress can only be physically founded; which is the very rationale behind the whole thing. Julio- Hide quoted text - - Show quoted text - IMHO, currently I don't believe the true geometry of space is even known. So at best, it seems that the physical world is a good starting point for considering likely canidates for useful geometries. The point I was making is for someone trying to learn math, as the top poster, worrying exactly what 'is' is at this stage maybe harmful to a learning math at deeper level. When you started learning math, what questions did you have that turned out to be not really that important? I had them. > Pardon me, but IMHO you risk here not to be thinking out of the box. I really don't know what you mean here. I'm willing to entertain any number of geometries or mathematical theories given that they meet some basic logical requirements, eg. consistency of axioms and etc. My requirements are pretty low as far as I'm concerned, given that the math is logically tight. Note: When I say arbitrary, I mean there is no mathematical reason to be considering one axiom over an another axiom; excluding logical considerations. It's given the real world will probably intrude into my considerations as to whether a mathematical construction has any useful purpose. This definitely wandering off topic now. Calan === Subject: Re: Is the empty set a number? > On Apr 8, 11:15æam, Julio Di Egidio > Pardon me, but IMHO you risk here not to be > thinking out of the box. > > I really don't know what you mean here. I'm willing > to entertain any > number of geometries or mathematical theories given > that they meet > some basic logical requirements, eg. consistency of > axioms and etc. > My requirements are pretty low as far as I'm > concerned, given that the > math is logically tight. > > Note: When I say arbitrary, I mean there is no > mathematical reason to > be considering one axiom over an another axiom; > excluding logical > considerations. It's given the real world will > probably intrude into > my considerations as to whether a mathematical > construction has any > useful purpose. Yes, and the point I am trying to make is: you have there just shown the very criterion to make our logic tighter without arbitrariness: purpose, that being a gain in applicability, so usefulness, etc. etc. > > This definitely wandering off topic now. On the edge indeed! :) Julio > > Calan === Subject: Re: Is the empty set a number? <9171313.1207760857492.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=7Ew-gAoAAACK7aClJ3SDozB1-FCoCyqu .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.1),gzip(gfe),gzip(gfe) > On Apr 8, 11:15æam, Julio Di Egidio > Pardon me, but IMHO you risk here not to be > thinking out of the box. I really don't know what you mean here. I'm willing > to entertain any > number of geometries or mathematical theories given > that they meet > some basic logical requirements, eg. consistency of > axioms and etc. > My requirements are pretty low as far as I'm > concerned, given that the > math is logically tight. Note: When I say arbitrary, I mean there is no > mathematical reason to > be considering one axiom over an another axiom; > excluding logical > considerations. æIt's given the real world will > probably intrude into > my considerations as to whether a mathematical > construction has any > useful purpose. Yes, and the point I am trying to make is: you have there just shown the very criterion to make our logic tighter without arbitrariness: purpose, that being a gain in applicability, so usefulness, etc. etc. This definitely wandering off topic now. On the edge indeed! :) Julio Calan- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - I think I basically agree with you on most points. However, strictly speaking, within restricted mathematical world, I believe that whatever purpose that a mathematical structure may have it doesn't change its ontological status, however, useful a branch of math maybe to us. It seems the flexability of an axiomatic system is ultimately what determines the useful of any set of axioms in the wider world. I can think of many instances where the lack of purpose of a branch of math was roundly critized only to be found it had some application not foreseen at its time of creation. thread. I have enjoyed reading them. Calan === Subject: Re: Is the empty set a number? > On Apr 9, 10:07æam, Julio Di Egidio > On Apr 8, 11:15æam, Julio Di Egidio > Pardon me, but IMHO you risk here not to be > thinking out of the box. I really don't know what you mean here. I'm > willing > to entertain any > number of geometries or mathematical theories > given > that they meet > some basic logical requirements, eg. consistency > of > axioms and etc. > My requirements are pretty low as far as I'm > concerned, given that the > math is logically tight. Note: When I say arbitrary, I mean there is no > mathematical reason to > be considering one axiom over an another axiom; > excluding logical > considerations. æIt's given the real world will > probably intrude into > my considerations as to whether a mathematical > construction has any > useful purpose. Yes, and the point I am trying to make is: you have > there just shown the very criterion to make our logic > tighter without arbitrariness: purpose, that > being a gain in applicability, so usefulness, etc. > etc. This definitely wandering off topic now. On the edge indeed! :) Julio Calan- Hide quoted text - - Show quoted text - > > I think I basically agree with you on most points. > However, strictly > speaking, within restricted mathematical world, I > believe that > whatever purpose that a mathematical structure may > have it doesn't > change its ontological status, however, useful a > branch of math maybe > to us. It seems the flexability of an axiomatic > system is ultimately > what determines the useful of any set of axioms in > the wider world. I > can think of many instances where the lack of > purpose of a branch of > math was roundly critized only to be found it had > some application not > foreseen at its time of creation. > > other posters on this > thread. > I have enjoyed reading them. > > Calan I want to thank you too, great discussion. You seem to put flexibility on top, and I can see the sense of it. Moreover your observation on the ontological status of numbers as independent from their purpose, that I'll have to digest, although I already agree. Julio === Hello everyone, 10April2008 ; Final 3rd I have followed the recent threads of K&K based themes. I am an amateur too and have often wondered why the Klauza-Klein 5D insight does not yield more depth and advocates. Ever ask yourself What is MASS and where is it? Well this theory asserts MASS is in a tiny volume that has its own laws. I claim to have made a significant discovery that relates mass and a fifth power law. I have loosely titled the theory : subSPACE ; The fifth dimensional domain -------------------------------------- subSPACE theory assumptions: 2. Normal spatial concepts of our macro scale do not apply to this subSPACE scale where mass resides; it's too distant. 3. The Fifth power LAW of subSPACE is mathematically related Since mass is already considered compacted, ANY valid mass relation/equation of a fifth power nature that encompasses most forms and range of mass would be a self evident proof and Klauza-Klein compatible. -------------------- Predictions; If this subSPACE thesis/model is correct, then 1 Its would be like a gamma ray burst , a sign literally proclaiming that mass/energy is a Five-Dimensional fact! this simple fifth power function; it would be indisputable. 4 It's Physics and Mathematics in a new one to one identity. The 5th dimension exists; eg. the near nuclear scales as universals. 5 A new mass/energy zone is predicted; It might offer insight into Dark matters! --------------------------------------------- Here are the first three 5D examples for the electron, proton and upper limit. e = (1)^5 was one standard e mass at 0.511 Mev Proton = (4.5)^5 at 1845.3 e masses or 943 Mev; a -0.5% fit. Max = (5.5)^5 at 5033 e masses or 2.6Bev Two more 5th power points emerge upon simple inspection of Kaon and Omega ranges. (4)^5 maps to the Kaon at 6% If (K + EtA)/2 mass is used , then its a 0.4% match. noted in this text as K' for the pairing. (5)^5 maps to the Omega at 4.8% So now there are five fits to a simple 5th power curve! e = (1)^5 was one standard e mass at 0.511 Mev K' as (Kaon+eta)/2 was 1024 e masses or 523 Mev at a -0.4% fit. Proton = (4.5)^5 at 1845.3 e masses or 943 Mev; a -0.5% fit. Omega at 5^5 was 3124 e masses or 1.6 Bev; at a 4.6% fit Max = (5.5)^5 or 5033 e masses at 2.6Bev ----------------------------- Next, the form of the function is changed to make a table of (n/2)^5 values from 1 to 11. The five points are now: (2/2)^5 (8/2)^5 (9/2)^5 (10/2)^5 (11/2)^5 e--------------K'------------P--------------Om----------Max 0 -0.4 -0.5 4.8 1 % match Further investigation seemed warranted! --------------------------------------------- CMF of 17.5 Mev; suggesting an internal mass grouping; a UNIT mass eg. Muon is ~6 UNITs , Pion is ~8 UNITs and Proton is ~54 UNITs Now there are six points, with five remaining unknowns. (2/2)^5 (4/2)^5 (8/2)^5 (9/2)^5 (10/2)^5 (11/2)^5 e------ CMF ----K'-----------P----------Om---------Max 0 7 -0.4 -0.5 4.8 1 % ----------------------- For further study, the remaining five 'n' values were separated into a (6/2, 7/2 & 5/2) group and 3/2 & 1/2 into another. -------------------------- Neutrino Match found Finally, consider the enigmatic (1/2)^5 value at e/32 masses or 16 Kev. It is an excellent fit to the Energy of anti-Neutrino measured in Tritium beta decay (0.1%) Here is current table of seven using ( n/2 )^5: n 1 2 3 8 9 10 11 ID eN e CMF K' P Om Max % err 0.1 0 7 -0.4 -0.5 4.8 1 What do you think so far, given these seven points? Especially given the (1/2)^5 to (11/2)^5 span; as the complete range of mass/energy, from a min of 16 Kev to max of 2.6 Bev. ---------------------------------------- This next Part will map the Muon and Pion at -2% & .7% fits; and present the full 1 to 11 mass range table, QED. and postulate Energy's five states. It also hypothesizes a Dark Matter candidate. The three remaining index values of 5,6 & 7 were the most difficult to match; as their 5th power values bracketed the muon and pion masses but were not close. If the (n/2)^5 function is thought of in a zero to twelve range, then six becomes the boundary between the upper and lower halves. a symmetric parity; n/2=6*2 Recall that the muon and pion are 6 & 8 CMF's and that the CMF is 32 e masses from (4/2)^5 . When the CMF is considered a UNIT, then the Muon is 6 minus a UNIT at 211 and the Pion is 6 plus a UNIT at 275 in e masses. Think of mass having an upper and a lower region with NOTHING in between -------the 5,6,& 7 as table mid range 5' Muon is 211 electron masses; (Mu = 243 -32); -2.1% 6' Table empty mid zone at (6/2)^5 at 243 e masses or ~7 CMFs. 7' Pion is 275 electron masses (Pi = 243 + 32); -0.7% An ad-hoc rule for the 6 slot was devised to split it into derived (5'=6'-1) and (7'=6'+1) slots that replace the values (5/2)^5 & (7/2)^5 , using the CMF as a UNIT of 32 e masses ------------------------------------------------------------------------ ---------------lets recap TABLE: Fifth power LAW of MASS as ( n/2 )^5; n=1 to 11. -----The upper five region from ( n/2 )^5 n Mass e mass error% ---------------------------------------------------------------------------- -------------- 11 MAX 5033 1 2.6 Bev 10 Om 3124 4.6 1.6 Bev 9 P 1845.3 -0.5 943 Mev 7' Pion 275 -0.7 (243+32 ) as 7'=6' plus UNIT /CMF upper ------ 6' Empty mid zone at n=(6*2)/2 --------------------------------------------------------- lower 5' Muon 211 -2.1 (243-32 ) as 5'=6' minus UNIT/CMF 4 CMF 32 Common Mass Factor at ~16Mev to 18Mev ; a UNIT of volume. 3 ? 7.6 3.88Mev ; an unknown zone ? Suggestions welcomed. 2 --e 1 NEGATIVE electron; table reference mass; 0.511 Mev ! 1 eN 1/32 0,1% Neutrino in Tritium Beta decay at 16 Kev; table MIN ----- the lower five region from ( n/2 )^5 ----------------------------------- Based on the 5th power law insights above, Energy seems to have Five states ! 1st as a Free gas-like state; Photons as E&M based and c limited. 2nd as Neutrinos 4th as a crystalline state; the CMF. 16 to 18 Mev volume zone around (4/2)^5 ). Hypothesis for the zone at ( 3/2 )^5 (3/2)^5 at 7.6 e masses at 3.88Mev reveals a new mass zone; Maybe a candidate for DARK Matter ? -------------------------------------------------------- Conclusions: 1 It's a flashing gamma ray burst , a sign literally proclaiming mass/energy is a 5D fact! it can not be disputed, given this definitive 5D power law! 3 It's Physics and Mathematics in a one to one identity. -- -------------------------------- Well, that's it; a mathematical 5th power function Comments are welcome as well as any ideas about the 3.88 Mev zone as it's the only unknown index in the 1 to 11 range. I think if anything is found around 3.88 Mev it would be an additional validation. The Fifth Dimension is conclusively revealed. send email to rdo.meara@pobox.com -- Yours truly RD [spring 2008 of ........ === posting-account=JpxxPAgAAAAgwzQIYqn4j6syK-YhOmcF Gecko/20070309 Firefox/2.0.0.3,gzip(gfe),gzip(gfe) > Ever ask yourself What is MASS and where is it? Well this theory asserts MASS is in a tiny volume > that has its own laws. > and that tiny volume is in Archimede Plutonium's pocket. === Subject: Re: New symbolic/numeric/dynamic/intuitive programming language wow... hello flaming thunder seems to be a pretty fantastic site. - Jason Stalnecker === Subject: Re: New symbolic/numeric/dynamic/intuitive programming language <30487164.1207855666326.JavaMail.jakarta@nitrogen.mathforum.org> <87lk3lqv12.fsf@phiwumbda.org> posting-account=qIx73woAAADp5gKOJOA4ARqKqxrzLb1s .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) > Isn't it keen when people with no previous posting history are so > taken with the overall awesomeness ofFlamingThunderthat they just > *have* to break their silence and gush about it? Yes. For example, Flaming Thunder's precise control of the rounding/truncation mode for high precision floating point arithmetic is already enabling one user to carry out research on stable points in chaotic maps that he was finding impossible to do in other computer languages -- but it was easy in Flaming Thunder. Are there any new features that you'd like to see? If you visit the News at http://www.flamingthunder.com/ you'll see that we've been adding something new every few days. === Subject: Re: New symbolic/numeric/dynamic/intuitive programming language <30487164.1207855666326.JavaMail.jakarta@nitrogen.mathforum.org> <87k5j4pnro.fsf@phiwumbda.org> posting-account=qIx73woAAADp5gKOJOA4ARqKqxrzLb1s .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) On Apr 11, 7:18æam, Tim Little made available in a future revision. Only in the Professional Edition. === Subject: Re: New symbolic/numeric/dynamic/intuitive programming language <30487164.1207855666326.JavaMail.jakarta@nitrogen.mathforum.org> <87k5j4pnro.fsf@phiwumbda.org> posting-account=qIx73woAAADp5gKOJOA4ARqKqxrzLb1s .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) > Is there any reason that your quoting software mangles quoted > material? It hasn't had any coffee yet this morning. === Subject: Re: New symbolic/numeric/dynamic/intuitive programming language posting-account=qIx73woAAADp5gKOJOA4ARqKqxrzLb1s .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) You can now set the approximation mode that Flaming Thunder uses for the real arithmetic operations +, -, * and /. The default mode is RoundToEven. Setting the number of real decimal digits also now words for +, -, * and / (see the News for 8 Apr 2008 at http://www.flamingthunder.com/ ). The 12 real approximation modes are: RoundToEven (probably the best for most applications) RoundToOdd RoundTowardNegativeInfinity RoundTowardPositiveInfinity RoundTowardZero TruncateToEven TruncateToOdd TruncateTowardNegativeInfinity TruncateTowardPositiveInfinity TruncateTowardZero To set the real approximation mode: Set RealApproximationMode to TruncateTowardZero. For example, suppose you want to see if 7 decimal digits are enough to sum the harmonic series 1/1 + 1/2 + ... 1/1_000_000. After all, 1_000_000 has only a little more than 6 digits. Let's suppose you also want to see the range of values you can expect under worst case conditions (truncating high and low instead of rounding). If you run the following program: Set Sum to 0.0. # Start with defaults: 30 digits, round to even. For n from 1 to 1_000_000 do set sum to sum + 1.0/n. Writeline 30 digits, round to even = , Sum. Set RealDecimalDigits to 7. # Use 7 digits for the rest. Set RealApproximationMode to TruncateTowardPositiveInfinity. Set Sum to 0.0. For n from 1 to 1_000_000 do set sum to sum + 1.0/n. Writeline 7 digits, truncate up = , Sum. Set RealApproximationMode to RoundToEven. Set Sum to 0.0. For n from 1 to 1_000_000 do set sum to sum + 1.0/n. Writeline 7 digits, round to even = , Sum. Set RealApproximationMode to TruncateTowardNegativeInfinity. Set Sum to 0.0. For n from 1 to 1_000_000 do set sum to sum + 1.0/n. Writeline 7 digits, truncate down = , Sum. it will write the following output: 30 digits, round to even = 14.3927267228657236313811274295 7 digits, truncate up = 21.61167 7 digits, round to even = 13.05426 7 digits, truncate down = 11.72242 So, you can conclude that 7 digits probably aren't enough. If you change 7 to 10 in the above program and rerun it, it will write: 30 digits, round to even = 14.3927267228657236313811274295 10 digits, truncate up = 14.39768079 10 digits, round to even = 14.39272306 10 digits, truncate down = 14.38779239 So, you can conclude that 10 digits are much better than 7. === Subject: Insanely difficult math question. What is the minimum number of times a fair die must be thrown for there to be at least an even chance that all scores appear at least once? === Subject: Re: Insanely difficult math question. <21427741.1207856448898.JavaMail.jakarta@nitrogen.mathforum.org>, > What is the minimum number of times a fair die must be thrown for there to be > at least an even chance that all scores appear at least once? > > Jason Stalnecker Let Ai be the event that in n throws, an i never occurs, i = 1, ..., 6. We want 1 - p(A1 U ... U A6). Inclusion-exclusion gives p(A1 U ... U A6) = sum p(Ai) - sum p(Ai n Aj) + sum p(Ai n Aj n Ak) - ... = 6*(5/6)^n - 15*(2/3)^n + 20*(1/2)^n - 15*(1/3)^n + 6*(1/6)^n. In Excel I'm getting n = 13 for the critical number. === Subject: Re: Insanely difficult math question. >What is the minimum number of times a fair die must be thrown for there to be at least an even chance that all scores appear at least once? For a fair 6 sided die, 13 throws are required. For an m-sided die, define f(n,k) to be the probability that, in _exactly_ n throws, all scores appear at least once, given that k scores have not yet been seen. Then it's easy to get the following recursive formula ... For nonnegative integers n,k with k <= m f(n,k) = 0 if either (n < k) or (k = 0 and n > 0) otherwise f(n,k) = (k/m)*f(n-1,k-1) + ((m-k)/m)*f(n-1,k) Can you explain the above recursive form? Think it through by considering what can happen from a given state, on a given throw. For the problem at hand, using m = 6, evaluate the sum f(6,6) + f(7,6) + f(8,6) + ... + f(N,6) stopping when the sum is greater or equal to 1/2. You find N = 13. quasi === Subject: Re: Insanely difficult math question. My previous reply needs some fixing. Here's a fixed version ... For a fair 6 sided die, 13 throws are required. For an m-sided die, define f(n,k) to be the probability that it takes exactly n throws for all scores appear at least once, given that k scores have not yet been seen. Then it's easy to get the following recursive formula ... For nonnegative integers n,k with k <= m f(0,0) = 1 f(n,0) = 0, if n > 0 f(n,k) = 0, if n < k otherwise f(n,k) = (k/m)*f(n-1,k-1) + ((m-k)/m)*f(n-1,k) Can you explain the above recursive form? Think it through by considering what can happen from a given state, on a given throw. For the problem at hand, using m = 6, evaluate the sum f(6,6) + f(7,6) + f(8,6) + ... + f(N,6) stopping when the sum is greater or equal to 1/2. You find N = 13. quasi === Subject: Re: Insanely difficult math question. posting-account=bSICGQkAAADSbkxAJ5uMxFegr4rp0Qig Gecko/20071115 Firefox/2.0.0.10,gzip(gfe),gzip(gfe) On Apr 10, 3:40 pm, Jason Stalnecker Look up the Coupon Collector Problem as a model. http://www.cs.tufts.edu/comp/250P/classpages/coupon.html - Randy === Subject: Re: No obama/mccain apologies -day 12 (still more racism) <1207526811_38690@news.usenet.com> posting-account=tQA_9wkAAAC8kqTdBlJnXYniA-oBKt-8 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Now (just for the effect). Explain this. Since you've failed to agree to your cheap red herring attempt having occurred, Explain away other Anti-Jewish Obama Campaign statements, like other Anti-Jewish claims from Obama campaign chairman McPeak, like this: Explain this away, kiddo: McPeak also charged Jews and Christian Zionists with dual-loyalties - www.israelnationalnews.com/News/News.aspx/125688 How about this: McPeak finds it more convenient to blame American Jewry and their perceived influence... - blogs.abcnews.com/politicalpunch/2008/03/obama-mcpeak- an.html This ?: The Republican Jewish Coalition (RJC) today called on Sen. Barack Obama to remove Gen. Merrill Tony McPeak as his military advisor - www.newsmax.com/insidecover/Jewish_Group_Wants_Obama_/2008/03/25/82924.html As stated. Obama/McCain surround themselves with racists and agree with them. === Subject: Re: looking for a nonnormal matrix > consider the ratio > where: > - y is a real n-dimensional vector; > - M is the orthogonal projector onto the orthogonal > complement of the column space of some nXk matrix X > with rank(X)=k > I am interested in maximizing r as y varies over R^n, > and for fixed A and M. > Conjecture: if r has a maximum at y=u for any M such > that Mu is not 0, then A satisfies i) and ii). > I would like to give an update to the previous post that may help to clarify my problem. I was trying to prove the above conjecture, where - A is a nonnegative and irreducible nxn matrix - i) is the following property: A and A^T have the same Perron eigenvector, to be denoted by u; - ii) is the following property: the matrix A+A^T has only two eigenvalues. Proof: Suppose A satisfies i) and ii) and let rho be the spectral radius of A. Then (2*rho,u) is an eigenpair of A+A^T. Calling s the other eigenvalue of A+A^T, we have A+A^T = 2*rho u u^T + s (I - u u^T) = s I_n + (2*rho-s) u u^T. Write M=Z^T Z for some (n-k)xk matrix Z such that Z Z^T=I_{n-k} (i.e. Z is semiorthogonal). Then, Z(A+A^T)Z^T = s I_{n-k} + (2*rho-s) Z u u^T Z^T. Now consider an (n-k)-dimensional vector v orthogonal to Zu. We have Z(A+A^T)Z^T v = s v, that is, the eigenspace of Z(A+A^T)Z^T associated to s is the orthogonal complement of Zu. But, since A+A^T is symmetric, Zu must be an eigenvector of Z(A+A^T)Z^T. By the Poincare' separation theorem the eigenvalue of Z(A+A^T)Z^T associated to Zu, to be denoted by t, must be larger than s. Hence, (u^T Z^T Z(A+A^T)Z^T Z u)/(u^T Z^T Z u)=t. Note that t is the maximum of r as y varies over R^n. It follows that the ratio r is maximized by y=u for any M such that Mu is not 0. === Subject: Re: inequalities posting-account=xLnn5AoAAAB6iFR7--UyDc2oxuzW22DA Gecko/20080311 Firefox/2.0.0.13,gzip(gfe),gzip(gfe) > Let P be a matrix with positive entries whose rows sums all equal 1. Let pi be a normalized left eigenvector of P, so pi' P = pi' and the > entries of pi sum to 1. Let D be a diagonal matrix with positive entries on the diagonal. Let mu be the vector PDe where e is a vector of ones. Let lambda be the largest eigenvalue of PD. 1) Under what conditions on P and D is there an inequality > relationship between lambda and pi' mu? 2) Under what conditions on P and D is there an inequality > relationship between log lambda and pi' log (mu)? [log (mu) is a vector whose entries are log mu_1, log mu_2, ... etc.] bump it up === Subject: Re: multidimensional scaling > dear statisticians, > .... You may have better luck finding statisticians in the news group. Ken Pledger. === Subject: Proving a sequence is dense, can yoou check my proof I'd like to have my proof checked, please: Suppose f (real valued)is continuous, periodic and non-constant on R. If its fundamental period p is irrational, then the sequence f(n), n=1,2,3..., is dense in f([0, p]). I already know the set A = {m*p + n | m integer, n positive integer} is dense in R. First, let's prove the following lemma: For every real x, there is a sequence x_k = m_k *p + n_k in A that converges to x and is such that n_k is strictly increasing. Proof: Since A is dense in R and every element of R is a limit point of R, every real number is a limit point of A. Hence, there exists a sequence w_i= m_i * p + n_i in A that converges to x and has its terms pairwise distinct. We claim w_i has a L-tail where the numbers n_i are pairwise distinct. Actually, since w_i converges, it is Cauchy. Hence, there is L such that i >= L and j >= L implies |w_i - w_j| = |(m_i - m_j) p + (n_i - n_j)| < p If n_i = n_j, then we get |m_i - m_j| < 1. Since m_i and m_j are integers, this implies m_i = m_j, so that w_i = w_j. But since this contradicts the assumption that the terms w_i are pairwise distinct, the claim is proved. Since the numbers n_i are positive integers and are pairwise distinct for i >= L, we can find a subsequence of the L-tail of w_i (so, a subsequence of w_i) where the numbers n_i form an strictly increasing sequence. This subsequence, as well as w_i, converges to x. By renumbering its terms, we get a sequence x_k = m_k *p + n_k as stated in the lemma. Now, let y be any number in f([0,p]). There is an x in [0, p] with y = f(x). According to the lemma, there's also a sequence w_k, as stated, that converges to x. Since f is continuous, w_k -> x, and p is a period of f, it follows that lim f(w_k) = lim f(m_k * p + n_k) = lim f(n_k) = f(x) = y. Since f(n_k) is a subsequence of f(n) and this holds for every y of f([0,p]), we conclude f(n) is dense in f([0,p]). I know there are better proofs based on complex numbers, but for now they are out of my reach. Sharon === Subject: Definition of Outer Measure? I have a problem in that I dont understand the definition of outer measure. We have definition 7.3.2 of outer measure here http://web01.shu.edu/projects/reals/integ/ on chapter 7. At the left, we have subchapter measures and the definition 7.3.2 of outer measure, which I am using: Of all possible countable coverings of E, we choose the one which has the least sum? I dont get it. For instance, in Example 7.3.3, first question: Question: Find the outer measure of the empty set O. Solution: Consider E = (-1/n, 1/n), which covers the empty set. It has length 2/n. Therefore m^*(O) < 2/n -> 0. My question: What in the definition allows us to let n -> infty? As I understand it, we should sum 2/n, n=natural numbers. Then we have a sum, 1/1 + 1/2 + 1/3 + ... which is not equal to the answer: 0. And how can that be the least possible sum, the infimum? We want the smallest sum, the infimum? Why do we choose (-1/n, 1/n)? Why not choose for instance (-1/(n*n), 1/(n*n)) which tends to 0 quicker than 1/n (i.e. it yields a smaller sum)? Hope someone can shed some light on this? === Subject: Re: Definition of Outer Measure? > Of all possible countable coverings of E, we choose the one which > has the least sum? I dont get it. For instance, in Example 7.3.3, > first question: Actually the term least is incorrect there: it should be infimum. > My question: What in the definition allows us to let n -> infty? The set (-1/n, 1/n) covers {0} for any n. The limit of any sequence is no less than the infimum of the set of values (though it may be greater). It's just using a concept (limit) that may be more familiar than infimum to illustrate the result. > As I understand it, we should sum 2/n, n=natural numbers. Then we > have a sum, 1/1 + 1/2 + 1/3 + ... which is not equal to the answer: > 0. No: We don't have a cover consisting of a sequence of sets, we have a sequence of covers consisting of one set each. The trivial one-term sum for each cover puts an upper bound on the outer measure. So for any epsilon > 0, the measure must be smaller because there exists n > 1/(2*epsilon). Since the outer measure cannot be less than 0, and infimum exists for any bounded set of real numbers, the only remaining possibility is that it equals zero. > And how can that be the least possible sum, the infimum? We want the > smallest sum, the infimum? Why do we choose (-1/n, 1/n)? The set of measures of this sequence are just one of many that have infimum zero. (-1/2^n, 1/n^2) would work such as well, though the algebra would be messier. > Why not choose for instance (-1/(n*n), 1/(n*n)) which tends to 0 > quicker than 1/n That also has infimum zero, but the speed which which it approaches zero is of no consequence at all. You could use that instead if you like; the result is the same. > Hope someone can shed some light on this? I hope I have. - Tim === Subject: Re: Definition of Outer Measure? > I have a problem in that I dont understand the definition of outer measure. > We have definition 7.3.2 of outer measure here > http://web01.shu.edu/projects/reals/integ/ > on chapter 7. At the left, we have subchapter measures and the definition 7.3.2 of outer measure, which I am using: > Of all possible countable coverings of E, we choose the one which has the least sum? I dont get it. I don't get it, either. Is that your paraphrase? I searched for least sum on that page and did not find it. What Definition 7.3.2 on that page says instead is that the outer measure is the *infimum* of all the sums taken over countable coverings of A. That's a very big difference, since the collection of sums is guaranteed to have an infimum, but may not have a least member. >For instance, in Example 7.3.3, first question: > Question: Find the outer measure of the empty set O. > Solution: Consider E = (-1/n, 1/n), which covers the empty set. It has length 2/n. Therefore m^*(O) < 2/n -> 0. > My question: What in the definition allows us to let n -> infty? The word *infimum* in the definition is what allows us to do that. For each n, we have a covering of the empty set consisting of the single interval (-1/n, 1/n). We then take the infimum over all such coverings, and obtain the result of 0. Notice, there are other possible coverings of the empty set that we have not considered here, but since the infimum over the given collection is 0, the infimum over *all* coverings certainly can't be any greater than that. It also can't be any less than that, since no sum of lengths can ever be negative. Hence, the infimum over all coverings of the empty set must be exactly 0. >As I understand it, we should sum 2/n, n=natural numbers. Then we have a sum, 1/1 + 1/2 + 1/3 + ... which is not equal to the answer: 0. The only sum involved here is the sum of the lengths in a given covering of the empty set. Since each of the coverings under consideration consists of just a single interval, each of the sums that results is a sum with exactly one term, namely S_n = 2/n. We then take the infimum of all the S_n, which is 0. > And how can that be the least possible sum, the infimum? We want the smallest sum, the infimum? Why do we choose (-1/n, 1/n)? Why not choose for instance (-1/(n*n), 1/(n*n)) which tends to 0 quicker than 1/n (i.e. it yields a smaller sum)? Speed of convergence doesn't matter. Feel free to use a different collection of coverings of the empty set. If your chosen collection yields an infimum of 0, then the reasoning outlined above still holds and shows that the outer measure of the empty set is 0. The infimum of the sums is not the same as the smallest sum. Proof: the given collection of sums does not have a smallest member, but it does have an infimum, also known as the greatest lower bound. It is a fundamental property of the real numbers that every collection of reals that is bounded below has a greatest lower bound. In this case, each of the (single-term) sums of lengths is bounded below by 0, and therefore a greatest lower bound exists. > Hope someone can shed some light on this? -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Definition of Outer Measure? > I have a problem in that I dont understand the > definition of outer measure. > > We have definition 7.3.2 of outer measure here > http://web01.shu.edu/projects/reals/integ/ > on chapter 7. At the left, we have subchapter > measures and the definition 7.3.2 of outer measure, > which I am using: > > Of all possible countable coverings of E, we choose > the one which has the least sum? I dont get it. For > instance, in Example 7.3.3, first question: No. The defition of outer measure is m(E) = infimum {L(G) | G is a countable open cover of E}, where L(G) is the total length of the open cover G. It's not the open cover with the least lenght, but the infimum of the set of lenghts. > > Question: Find the outer measure of the empty set > O. > Solution: Consider E = (-1/n, 1/n), which covers the > empty set. It has length 2/n. Therefore m^*(O) < 2/n > -> 0. > > > My question: What in the definition allows us to let > n -> infty? As I understand it, we should sum 2/n, > n=natural numbers. Then we have a sum, 1/1 + 1/2 + > 1/3 + ... which is not equal to the answer: 0. No, this is not the case. For each n, the set {(-1/n, 1/n)} can be seen as an open cover of the the empty set composed of the single interval (-1/n, 1/n). Given eps >0, you set n > 2/eps and the lenght of the open cover {(-1/n, 1/n)} is 2n < eps. So, for every eps >0 you can find a countable (in the case, finite) open cover G of the empty set with L(G) < eps. It follows the infimum of the set of the lenghts of all countable open covers of the empty set is 0. > > And how can that be the least possible sum, the > infimum? We want the smallest sum, the infimum? Why > do we choose (-1/n, 1/n)? Why not choose for instance > (-1/(n*n), 1/(n*n)) which tends to 0 quicker than 1/n > (i.e. it yields a smaller sum)? You can do it, but it doesn't matter. Once you produce any sequence of countable open covers such that the infimum of their lenghts is 0, you proved the outer measure is 0. > > Hope someone can shed some light on this? I hope this helps a bit Artur === Subject: Re: Definition of Outer Measure? > And how can that be the least possible sum, the > infimum? We want the smallest sum, the infimum? > Why do we choose (-1/n, 1/n)? Why not choose for > instance (-1/(n*n), 1/(n*n)) which tends to 0 quicker > than 1/n (i.e. it yields a smaller sum)? The issue is not how quick you can tend to zero, but rather what is the infimum of all the possible sums. In the case of the empty set, any real positive real number occurs as such a sum [proof: Let r > 0. Then (-r/2, r/2) covers the empty set and has length equal to r], in many ways in fact, and since any such sum has to also be a positive real number, the set we're supposed to take the infimum of in this case is the set of all positive real numbers: inf{t: t > 0} This infimum is equal to 0. To emphasize a point that you seemed to missunderstand, the infimum is over a particular set of positive real numbers. The sequences used in the example you cited are just a means to find this infimum. Dave L. Renfro === Subject: Re: Definition of Outer Measure? Orvar Korvar a .8ecrit : > I have a problem in that I dont understand the definition of outer measure. > > We have definition 7.3.2 of outer measure here > http://web01.shu.edu/projects/reals/integ/ > on chapter 7. At the left, we have subchapter measures and the definition 7.3.2 of outer measure, which I am using: > > Of all possible countable coverings of E, we choose the one which has the least sum? I dont get it. For instance, in Example 7.3.3, first question: > > Question: Find the outer measure of the empty set O. > Solution: Consider E = (-1/n, 1/n), which covers the empty set. It has length 2/n. Therefore m^*(O) < 2/n -> 0. > > > My question: What in the definition allows us to let n -> infty? As I understand it, we should sum 2/n, n=natural numbers. Then we have a sum, 1/1 + 1/2 + 1/3 + ... which is not equal to the answer: 0. You are simply confused with the indices --- Consider, for k integer >= 1 fixed, the covering of A={} by union A_n with for n=1, A_1 = E_k = (-1/k,1/k), for n>=2, A_n = {} = empty set. We have sum_n measure(A_n) = 2/k. This means that outer_measure(A) <= 2/k for all k >= 1 and thus ... HTH, Best, Amities, Olivier