An uncountable countable set
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From: Mike Kelly on 19 Sep 2006 06:50 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>Mike Kelly wrote: > >>>> > >>>>>Han de Bruijn wrote: > >>>>> > >>>>>>All naturals do not exist. What is "all"? > >>>>> > >>>>>Huh? So some naturals don't exist? What does that mean? How can > >>>>>something that doesn't exist be a natural number? > >>>> > >>>>"All naturals" is undefined, void of meaning. Got it? > >>> > >>>Not really, no. If something is true for "All naturals" it is true for > >>>any object which is a natural number. The set of all naturals is the > >>>set which has as an element every object which is a natural number and > >>>no element which is not a natural number. > >>> > >>>I suppose you're just going to respond "but there is no such set!" - > >>>presupposing your conclusion; that there are no "completed infinities". > >>>Vigorous assertion is so convincing. > >> > >>So there is no lack of understanding on your part. Good! > > > > So you admit you don't actually have an argument beyond assertion? At > > least mathematicians recognise that the "existence of completed > > infinities" is not provable one way or the other. > > No argument beyond assertion AND (empirical) scientific evidence: > > http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#oo What bearing does empirical evidence have on what is mathematically meaningful? -- mike.
From: mueckenh on 19 Sep 2006 07:08 Mike Kelly schrieb: > > Any set that can be established is a finite set. > > Why? Look: If aleph_0 were a number larger than any natural number, then for any natural number n we had n < aleph_0. "For all" means: even in the limit. So lim [n-->oo] n/aleph_0 < 1 If aleph_0 counted the numbers, for instance the even naturals, then we had for all of them lim [n-->oo] |{2,4,6,...,2n}| = aleph_0. This would yield lim [n-->oo] (2n/|{2,4,6,...,2n}|) < 1 But we have lim [n-->oo] (2n/|{2,4,6,...,2n}|) = 2 Therefore aleph_0 does not exist as a number which could be compared with other numbers. > > > Hence, the probability to select a number divisible by 3 is 1/3 or very very close to 1/3. > > >From finite sets of consecutive naturals when selecting with a uniform > distribution, sure. But you don't accept that there is a set of "all" > natural numbers so what does it mean to you to select at random from > "all" naturals? YOU should know it, if they all exist for you. Take all of them and then select one without looking. > > > > >Otherwise the > > > > limit of the sequence 1/n might be 100. Nobody could prove that false. > > > > > > Babble. > > > > 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. > > Prove that to be the case without using any definition of what a series > is and without any axioms. Archimedes did so when exhausting the area of the parabola. In decimal notation 2 + 2 = 4, and in any system we have II and II = IIII. For self-evident truths you don't need axioms. Only if you want to establish uncertain things like "There exist a set which contains O and with a also {a}" then axioms may be required. Don't misunderstand me: I do not oppose the principle of induction but the phrase "there exists" which suggests the existence of the completed set. Regards, WM
From: mueckenh on 19 Sep 2006 07:13 Mike Kelly schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Mike Kelly schrieb: > > > > > It is meaningful to say that a natural drawn uniformly at random from a > > > set of consecutive naturals 1 thru 3n has a 1/3 probabaility of being > > > divisible by 3. Nobody disputes this. But talking about the probability > > > of "a natural" being divisible by 3 implies a uniform distribution over > > > the naturals. Such a thing does not exist. > > > > Talking about sinx / x for x --> 0 does not imply the existence of sin0 > > / 0. Neither does the result 1/3 imply the distribution for a realy > > infinite set f naturals. There is no real (actual, finished) infinity, > > neither in physics > > Ok. > > >nor in mathematics. > > Why? Because all we are and all we think with (brain, neurons, currents, loads, and ideas, letters, words, pictures, i.e., hardware and software) and all we think and all we think we are thinking: all that is physics. Regards, WM
From: Han.deBruijn on 19 Sep 2006 07:13 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. 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. 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! 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
From: Han de Bruijn on 19 Sep 2006 07:21
Mike Kelly wrote: > Han de Bruijn wrote: > >>Mike Kelly wrote: >> >>>Han de Bruijn wrote: >>> >>>>Mike Kelly wrote: >>>> >>>>>Han de Bruijn wrote: >>>>> >>>>>>Mike Kelly wrote: >>>>>> >>>>>>>Han de Bruijn wrote: >>>>>>> >>>>>>>>Mike Kelly wrote in response to Tony Orlow: >>>>>>>> >>>>>>>>>*sigh*. Probabilities are *standard* real numbers between 0 and 1. >>>>>>>> >>>>>>>>Yes. And infinitesimals are *standard* real numbers in engineering. >>>>>>> >>>>>>>Engineering is not mathematics. It uses mathematical results. >>>>>> >>>>>>Sure. And moslems are not praying. Only roman catholics do. >>>>> >>>>>Bizarre and borderline offensive analogy. Engineering isn't >>>>>mathematics, anymore than accounting is mathematics. They both *use* >>>>>mathematics. >>>> >>>>Wrong. They both *create* mathematics as well. I know, because I've been >>>>active in both engineering and adminstration. >>> >>>But not in mathematics, evidently. So how do you tell the application >>>of mathematics from mathematics itself, then? >> >>Good question! I find it impossible to distinguish them, while it seems >>that you can do it. Unambiguously ? Please tell me how, because I don't >>understand. > > Mathematics is the derivation of theorems from axioms. The application > of mathematics is using the theorems of some mathematical theory to > solve problems that the theory can model. Then I am right and your so-called "applications" are a form of active mathematics. Mathematicians simply can not and do not come up with all the theorems most modern applications are in the need of. So we develop them ourselves. Whether this situation is desirable is another question. >>>>>>>>That's why infinitesimal probabilities will become feasible as soon >>>>>>>>as mathematics becomes a science which is compliant with engineering. >>>>>>> >>>>>>>Mathematics is not a science. What exactly would it mean for it to >>>>>>>"become a science compliant with engineering"? >>>>>> >>>>>>Ah! Mathematics is not a science. So mathematics is not serious at all! >>>>>>Why didn't you tell me this before? >>>>> >>>>>How do you get from "mathematics is not a science" to "mathematics is >>>>>not serious"? Time to brush up on your English, Han. >>>> >>>>How can mathematics be serious, if it is not scientific? >>> >>>Definately a translation issue. Can I expect a lecture from you on the >>>proper meaning of the word "science", now? >> >>Not now. But _you_ started saying that "Mathematics is not a science". >>How can you be so sure, if you need an explanation from me? > > I don't need an explanation from you. You really can't read English, > can you? You can't read Dutch, can you? Han de Bruijn |