An uncountable countable set
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From: Virgil on 19 Sep 2006 02:40 In article <70eb8$450e4cce$82a1e228$14803(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <cf41b$450a799d$82a1e228$10666(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>But, fortunately, reality is more simple than this. Every equality is > >>an equivalence relation. And every equivalence relation is an equality. > >>So the bijection between tally marks or collected pebbles with counted > >>objects means, indeed, that tally marks and moving pebbles ARE numbers. > > > > Not in mathematics. > > Not in _your_ mathematics. > > Han de Bruijn Where, in mathematics or anywhere else, is a pebble indistinguishable from a nick in a stick? Of one can distinguish between them, then they are not the same thing.
From: Virgil on 19 Sep 2006 02:41 In article <cb4c3$450e672a$82a1e228$20090(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <450c3c65(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >>What is the average value of the reals in [0,1]? > > > > By what definition of average? > > > > There are a whole bunch of such definitions. > > > > Note that many such definitions only apply to finite sets of numbers, > > and thus won't work. > > > > Also, since the set [0,1] is invariant under x --> x^n, its average > > should be also. > > Objection! One such definition is: > > Average(x) = integral(0,1) x dx / integral(0,1) dx = 1/2 > > And, do you suggest that: > > Average(x) = integral(0,1) x^n . x dx / integral(0,1) x^n dx > = (n+1)/(n+2) = 1/2 for arbitrary n ? > > That's a weird suggestion. No? > > Han de Bruijn You might be able to object, but TO would not have, at least without help.
From: Virgil on 19 Sep 2006 02:43 In article <48cb1$450e6a48$82a1e228$20803(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > >>Does 1/aleph_0 lie within the real interval [0,1]? > > > > AS "1/aleph_0" is not a real number at all, and real intervals contain > > nothing other than real numbers, "1/aleph_0" does not lie within ANY > > real interval. > > Indeed. But if aleph_0 is considered as a finite quantity in infinite > disguise, then I can nevertheless attach a meaning to TO's babblings. Only within your own babblings.
From: Virgil on 19 Sep 2006 02:44 In article <c74ab$450e6bf9$82a1e228$21037(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > It doesn't matter what TO tries to say about it, it still will not make > > either aleph_0 into a real number, nor any alleged reciprocal of aleph_0 > > into a real number. > > TO's aleph_0 is obviously different from yours. What he means is that > the reciprocal of an infinitely large natural number is an infinitely > small real number, as every engineer knows. Infinitely large/small is > just "very" large/small in engineering terms, but I know that is very > much "undefined" in mainstream mathematics. (I know, I know ...) > > Han de Bruijn If To were to say it within a context of engineering, that would be one thing, but he says it in the context of set theory.
From: Virgil on 19 Sep 2006 02:47
In article <d7487$450e76b4$82a1e228$24975(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Mike Kelly wrote [ OK, let's keep the quotes intact this time ]: > > > If you don't accept the existence of a set of natural numbers then you > > don't accept the set theory that probability theory is based upon and > > you haven't suggested an alternative. Indeed, it seems somewhat odd to > > complain about the conclusion of a theorem discussing an object you > > don't accept even exists. > > The infinitary part of set theory that underpins probability theory is > IMHO the only problem. If you do not accept infinite sets at all, you can hardly insist on any sort of probability density functions being defined on them. |