An uncountable countable set
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From: Virgil on 29 Sep 2006 15:21 In article <e4ca4$451cd0dd$82a1e228$14108(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <1159437062.473100.294820(a)k70g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >>Virgil schrieb: > >> > >>>Several sets may all have the common property of being pairwise > >>>bijectable, but if any of their members are distinguishable from those > >>>of another set then the sets are equally distinguishable. > >> > >>Each one of the sets expresses, represents, and *is* the same > >>(cardinal) number. > > > > Then one apple and one orange are the same because they have the same > > cardinality. > > In _that_ respect, with respect to counting: definitely, yes! But not necessarily in any other respect whatsoever, so that to say an apple is an orange or an orange is an apple, as some have been saying, is foolishly wrong.
From: Virgil on 29 Sep 2006 15:23 In article <2b79d$451cd38b$82a1e228$17494(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <76b59$451ba0bd$82a1e228$18077(a)news2.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>mueckenh(a)rz.fh-augsburg.de wrote: > >> > >>>Virgil schrieb: > >> > >>>>>>You stated that you needed counting to determine the successor. That is > >>>>>>false. The successor is defined without any reference to counting. > >>>>> > >>>>>The successor function *is* counting (+1). > >>>> > >>>>Not to those who can't count. Successorship does not require numbers, it > >>>>only requires "next". > >>> > >>>How far would those who cannot count be able to find "the next"? > >> > >>And how do you distinguish "the next" from something previous? > > > > By pointing at them separately. > > > >>This is > >>not a joke. Many young children don't find it trivial that you shouldn't > >>count a thing twice. > > > > But they are much less prone to mistaking who has more marbles, or > > whatever, which argues that injection, surjection and bijection are more > > basic than counting. > > Have two bags with say a hundred marbles in it and _make_ the bijection. > I wish you good luck. And, BTW, I would like to have a computer program > which does the job, properly. Video circuit attached. > The age at which children would be able to compare bags of about n marbles successfully ins an increasing function of the age of the children. There is an age in which they could not even compare empty bags.
From: Virgil on 29 Sep 2006 15:26 In article <38a22$451cd683$82a1e228$19346(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > stephen(a)nomail.com wrote: > > > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > >>Virgil wrote: > > > >>>In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, > >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>> > >>>>Randy Poe wrote, about the Balls in a Vase problem: > >>>> > >>>>>It definitely empties, since every ball you put in is > >>>>>later taken out. > >>>> > >>>>And _that_ individual calls himself a physicist? > >>> > >>>Does Han claim that there is any ball put in that is not taken out? > > > >>Nonsense question. Noon doesn't exist in this problem. > > > > Yes it is a nonsense question, in the sense > > that it is non-physical. You cannot actually perform > > the "experiment". Just as choosing a number uniformly > > from the set of all naturals is a non-physical nonsense > > question. You cannot perform that experiment either. > > But you _can_ do it at any time _before_ noon. There is no limit > of the number of balls before noon which converges at noon. > > But you _can_ do it with any finite contiguous set of naturals. > Then you find floor(n/a)/n and with limit(n -> oo) find 1/a . But that does not define a uniform distribution of a countably infinite set of naturals as that would require that enough 0's will add up to 1. > > Han de Bruijn
From: Virgil on 29 Sep 2006 15:32 In article <451d5d29$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <1159438112.240001.268540(a)m7g2000cwm.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >> Dik T. Winter schrieb: > >> > >>> > The successor function *is* counting (+1). > >>> > >>> Wrong. > >> After a while you will have run out of the predefined successor, > >> unavoidably. > > > > If that were ever to happen, one would have discovered a largest > > possible number. But it does not ever happen, because for every set x > > there is a set UNION(x,{x}) which is its successor. > > > > I believe Wolfgang is saying that, once you run out of the starting > known successive symbols of your language, your alphabet, you then have > to employ an actual number system, using those elements recursively. > Since alphabets are generally finite, you can never represent "infinite" > quantities, in terms of string length. One does not need to, as every finite natural is representable by a finite string. > > As you know, I like to represent specific infinite quantities using > finite strings, but of course it's only a countable set of infinite > numbers, since the infinite sequences are defined using repeating > patterns, making them rational fractions of declared infinities. They're > T-riffic! :) More like Horrific! All the finite strings are already in use for finite naturals, so how does TO propose to re-use them for infinite numbers? Print them in a different color? Won't work in NGs.
From: Virgil on 29 Sep 2006 15:37
In article <451d5e15(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Han de Bruijn wrote: > >>> Virgil wrote: > >>> > >>>> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, > >>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>>> > >>>>> Randy Poe wrote, about the Balls in a Vase problem: > >>>>> > >>>>>> It definitely empties, since every ball you put in is > >>>>>> later taken out. > >>>>> And _that_ individual calls himself a physicist? > >>>> Does Han claim that there is any ball put in that is not taken out? > >>> Nonsense question. Noon doesn't exist in this problem. > >>> > >>> Han de Bruijn > >>> > >> That's the question I am trying to pin down. If noon exists, that's when > >> the vase supposedly empties, > > > > Why does the existence of noon imply there is a time > > which is the last time before noon? > > > > It doesn't. > > > > - Randy > > > > I never said it did. When did I say that? I will offer this simple > logical argument. If the vase ever became empty, it would be because one > ball was removed, as per the gedanken, but 10 balls would have been > inserted immediately beforehand. The vase would therefore have had to > contain -9 balls, which I'm afraid is simply impossible. Don't you? It's > a ridiculous set-theoretical result. So is TO's conclusion that there ought to be infinitely many naturally numbered balls in the vase for which he can not find the number of any one of them. Let us consider a slightly modified experiment in which as each ball is removed from the vase, it is put into an initially empty urn. Now at or after noon, ball 1 is in the urn (and not in the vase). Furthermore for every n-marked ball in the urn, ball n+1 is also in the urn. Thus, by induction, EVERY naturally numbered ball is in the urn (and not in the vase). So which balls are still in the vase and not in the urn, TO? |