An uncountable countable set
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From: Mike Kelly on 19 Sep 2006 11:00 Han de Bruijn wrote: > Mike Kelly wrote: > > > What is "active mathematics"? > > > I'm not really seeing the point in this line of discussion. In what way > > does any of this dispute that probabilities are not infinitesimal? > > > I'm not attempting to argue in Dutch, am I? > > Your attitude is quite constructive, huh? Your habit of snipping all context is quite annoying, huh? And still the question remains : what does all this have to do with probabilities not being infinitesimal? -- mike.
From: Mike Kelly on 19 Sep 2006 12:12 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Mike Kelly wrote [ dishonestly snipping context, as usual ] : > > > Han de Bruijn wrote: > > > > > > Mike Kelly wrote: > > > > > > > > Your claim was that *standard* set theory + calculus contradicts > > > > *standard* probability theory. This is untrue. Do you admit it? > > > > > > Of course not. > > > > Of course you don't admit it? Even though you're *wrong*? > > I'm not wrong. We'll see. > Consider the Riemann sum corresponding with the integral from 0 to 1 > over the function f(x) = 1. It is: sum(n=1,n) 1.1/n = n.1/n = 1. Imprecise but accurate enough. > A picture says more than a thousand words: > > http://hdebruijn.soo.dto.tudelft.nl/jaar2006/calculus.jpg > > Now consider the set {1, 2, 3, ... , n}. Check that the sum of all > elementary probabilities is one: 1/n + 1/n + ... + 1/n = 1/n.n = 1 . > > Again, a picture says more than a thousand words: > > http://hdebruijn.soo.dto.tudelft.nl/jaar2006/probable.jpg > > But hey! That is quite a different picture! Careful... a distribution on a set of natural numbers is discrete. But let's see the meat of your argument. > Maybe so. But the two formulas are IDENTICAL: 1/n.n = 1 . > > Meaning either that the Riemann sum of the integral(0,1) dx = 1 OR > that the Kolgomorov axiom for the sum of elementary probabilities > is valid. In a sensible mathematics, the OR is inclusive. Because > nobody can tell whether 1/n.n = 1 is the Riemann or the Kolgomorov > summation. This remains so for arbitrary large n. > > But now the miracle happens. Let n -> oo. > > Then the Riemann sum converges to the integral(0,1) dx. No problem. > > But the Kolgomorov sum, which is EXACTLY THE SAME FORMULA, suddenly > converges to ... nothing?! > > Well, that may be in your head, but not in any sensible mathematics. > > Han de Bruijn So... for the function f(x) = 1 between 0 and 1 we have a sequence of Reimann sums 1/1, 2/2, 3/3, 4/4, ... = 1, 1, 1, 1... Clearly this sequence converges to 1, so the Reimann integral of f(x) = 1 between 0 and 1 is 1. Fine and dandy, so far. Now we have a sequence of sets.. {0}, {0,1}, {0,1,2}, {0,1,2,3} .... and for each we can define a uniform probability distribution to choose one element of the set. The limit of this sequence of sets is N. Now what? We assume that we can define a uniform probability distribution on N to choose a natural number? Because this property holds for each element of the sequence? But it is not the case that the limit of a sequence need have all the properties of the elements of a sequence. So... what is your argument? Oh, so you want to have a corresponding sequence of "Komologrov sums" which are the summation of the elementary probabilities? 1/1, 2/2, 3/3, 4/4, ... = 1, 1, 1, 1... You are quire wrong to claim this sequence has no limit. Clearly the limit of this sequence is 1. So now we have two sequences here. The sets, and the corresponding "Komologrov sums". {0), {0, 1}, {0, 1, 2}, {0, 1, 2, 3} ... 1, 1 , 1 , 1... Aaaand... so what? You want to conclude that since the limit of the sequence of sets is N and the limit of the corresponding "Komologrov sums" is 1 that there is some correspondence between the two limits and that therefore there is a uniform probabiltiy distribution on N? Based on what? The limit of a sequence need not have a property that each of its elements has. -- mike.
From: MoeBlee on 19 Sep 2006 12:33 Han de Bruijn wrote: > MoeBlee wrote: > > > Han de Bruijn wrote: > > > >>MoeBlee wrote: > >> > >>>Tony Orlow wrote: > >>> > >>>>Unbounded but finite may > >>>>be considered potentially, but not actually, infinite. > >>> > >>>That will be jiffy, once you give axioms and/or our definitions for > >>>'potentially infinite' and 'actual infinite'. Until then, it's pure > >>>handwaving. > >> > >>Come on, Moeblee, don't be silly! Let Google be your friend! > > > > I'm well aware of the notions of 'potential infinity' and 'actual > > infinity' that go back through at least a couple thousand years of > > philosophy and philosophy of mathematics. But to use those notions in > > Here comes: > > > an axiomatic mathematics requires either defining the terms > > 'potentially infinite' and 'actually infinite' in the axiomatic theory > > or taking them as primitive and giving axioms for them in the axiomatic > > theory. So far, that has not been done in this thread or in any other > > thread I've happened to read. > > Now look what WM says, elsewhere in this thread: > > > No. Just this is the point! The series 1 + 1/2 + 1/4 + ... is 2 (or at > > least as close to 2 as we like), not by definition and not by any > > axiom, but by rational thought. And the same kind of extrapolation is > > appropriate if we investigate the infinite, be it the sequence 1/n or > > the "bijection" N <--> Q. > > Let's repeat the essential phrase in capital letters: > > NOT BY ANY AXIOM, BUT BY RATIONAL THOUGHT. I responded to Orlow's claim that there are unbounded but finite sets; I was not in response to, and I made no mention of, WM's remarks that you just quoted. MoeBlee
From: MoeBlee on 19 Sep 2006 12:51 MoeBlee wrote: > Han de Bruijn wrote: > > MoeBlee wrote: > > > > > Han de Bruijn wrote: > > > > > >>MoeBlee wrote: > > >> > > >>>Tony Orlow wrote: > > >>> > > >>>>Unbounded but finite may > > >>>>be considered potentially, but not actually, infinite. > > >>> > > >>>That will be jiffy, once you give axioms and/or our definitions for > > >>>'potentially infinite' and 'actual infinite'. Until then, it's pure > > >>>handwaving. > > >> > > >>Come on, Moeblee, don't be silly! Let Google be your friend! > > > > > > I'm well aware of the notions of 'potential infinity' and 'actual > > > infinity' that go back through at least a couple thousand years of > > > philosophy and philosophy of mathematics. But to use those notions in > > > > Here comes: > > > > > an axiomatic mathematics requires either defining the terms > > > 'potentially infinite' and 'actually infinite' in the axiomatic theory > > > or taking them as primitive and giving axioms for them in the axiomatic > > > theory. So far, that has not been done in this thread or in any other > > > thread I've happened to read. > > > > Now look what WM says, elsewhere in this thread: > > > > > No. Just this is the point! The series 1 + 1/2 + 1/4 + ... is 2 (or at > > > least as close to 2 as we like), not by definition and not by any > > > axiom, but by rational thought. And the same kind of extrapolation is > > > appropriate if we investigate the infinite, be it the sequence 1/n or > > > the "bijection" N <--> Q. > > > > Let's repeat the essential phrase in capital letters: > > > > NOT BY ANY AXIOM, BUT BY RATIONAL THOUGHT. > > I responded to Orlow's claim that there are unbounded but finite sets; > I was not in response to, and I made no mention of, WM's remarks that > you just quoted. P.S. My point is that I said that I have not yet read definitions of 'potentially infinite' and 'actually infinite' in an axiomatized theory nor have I read of them as primitives with axioms for them. Then you mention that WM claims that rational thought alone, without axioms or definitions, shows that the terms of a series get arbitrarily close to some value. Whatever the merit of that claim, it doesn't meet my challenge, which is to provide an axiomatization for 'potentially infinite' and 'actually infinite'. MoeBlee or axiomatization of 'po
From: Tony Orlow on 19 Sep 2006 13:17
MoeBlee wrote: > MoeBlee wrote: >> Han de Bruijn wrote: >>> MoeBlee wrote: >>> >>>> Han de Bruijn wrote: >>>> >>>>> MoeBlee wrote: >>>>> >>>>>> Tony Orlow wrote: >>>>>> >>>>>>> Unbounded but finite may >>>>>>> be considered potentially, but not actually, infinite. >>>>>> That will be jiffy, once you give axioms and/or our definitions for >>>>>> 'potentially infinite' and 'actual infinite'. Until then, it's pure >>>>>> handwaving. >>>>> Come on, Moeblee, don't be silly! Let Google be your friend! >>>> I'm well aware of the notions of 'potential infinity' and 'actual >>>> infinity' that go back through at least a couple thousand years of >>>> philosophy and philosophy of mathematics. But to use those notions in >>> Here comes: >>> >>>> an axiomatic mathematics requires either defining the terms >>>> 'potentially infinite' and 'actually infinite' in the axiomatic theory >>>> or taking them as primitive and giving axioms for them in the axiomatic >>>> theory. So far, that has not been done in this thread or in any other >>>> thread I've happened to read. >>> Now look what WM says, elsewhere in this thread: >>> >>>> No. Just this is the point! The series 1 + 1/2 + 1/4 + ... is 2 (or at >>>> least as close to 2 as we like), not by definition and not by any >>>> axiom, but by rational thought. And the same kind of extrapolation is >>>> appropriate if we investigate the infinite, be it the sequence 1/n or >>>> the "bijection" N <--> Q. >>> Let's repeat the essential phrase in capital letters: >>> >>> NOT BY ANY AXIOM, BUT BY RATIONAL THOUGHT. >> I responded to Orlow's claim that there are unbounded but finite sets; >> I was not in response to, and I made no mention of, WM's remarks that >> you just quoted. > > P.S. My point is that I said that I have not yet read definitions of > 'potentially infinite' and 'actually infinite' in an axiomatized theory > nor have I read of them as primitives with axioms for them. Then you > mention that WM claims that rational thought alone, without axioms or > definitions, shows that the terms of a series get arbitrarily close to > some value. Whatever the merit of that claim, it doesn't meet my > challenge, which is to provide an axiomatization for 'potentially > infinite' and 'actually infinite'. > > MoeBlee > > or axiomatization of 'po > Given N is the standard naturals: Actually infinite(s) = E seS A neN index(s)>n Potentially infinite = A seS index(s)eN ^ ~E neN index(s)<>n |