An uncountable countable set
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From: Virgil on 2 Oct 2006 04:20 In article <c7f7b$4520c367$82a1e228$22987(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <c87e0$451cc5b4$82a1e228$4275(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>MoeBlee wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>It's a priorities issue. Do axioms have to dictate what constructivism > >>>>should be like? Should constructivism be tailored to the objectives of > >>>>axiomatics? I think not. > >>> > >>>Fine, but if you don't give a formal system, then your mathematical > >>>arguments are not subject to the objectivity of evaluation that > >>>arguments backed up by formal systems are subject to. > >> > >>Exactly! Constructivism is not Formalism. > > > > Do constructivists have any statements which they accept as true without > > proof? > > If not how do they prove anything from nothing? > > If so, then aren't those things they accept equivalent to axioms. > > Constructively valid proof. Intuition precedes axioms. > A mathematician > is like an architect who builds his mathematics. So that those who accuse formalists of building on sand merely build on a different, less visible, kind of sand.
From: Virgil on 2 Oct 2006 04:21 In article <21b4$4520c439$82a1e228$22987(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <51de2$451cc7e9$82a1e228$6256(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>Virgil wrote: > >> > >>>In article <2e658$451b78ef$82a1e228$7519(a)news1.tudelft.nl>, > >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>> > >>>>Tony Orlow wrote: > >>>> > >>>>>MoeBlee wrote: > >>>>> > >>>>>>Tony Orlow wrote: > >>>>>> > >>>>>>>Constructivism and Axiomatism are two sides of a coin. They can be > >>>>>>>reconciled in larger framework, I think. > >>>>>> > >>>>>>I don't know what your definition of 'axiomatism' is, but there are > >>>>>>axiomatic systems for constructive mathematics. > >>>>> > >>>>>I dunno. I was responding to Han's comment. I think he means > >>>>>constructive concepts vs. axiomatic declarations. > >>>> > >>>>It's a priorities issue. Do axioms have to dictate what constructivism > >>>>should be like? Should constructivism be tailored to the objectives of > >>>>axiomatics? I think not. > >>> > >>>But if you cannot clearly state what you are assuming/accepting as true, > >>>all you have is a morass of ambiguity. > >> > >>Ambiguity does not necessarily comprise a morass. > > > > How do constructivists deduce new things if there is nothing that they > > can say is true to start with? > > Constructivism starts with counting. All the rest is architecture. Then they are assuming things that they do not make explicit.
From: Virgil on 2 Oct 2006 04:23 In article <65e14$4520c63d$82a1e228$23833(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > 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. > > Why? Give me one piece of fruit. I don't care whether it is an orange or > an apple .. A bunch of grapes is one piece.
From: Virgil on 2 Oct 2006 04:26 In article <bfe8f$4520c878$82a1e228$24731(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > 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. > > A long time ago, I was reading a pictures book with one of my children. > There was a flower on page one, a bear on page two, a fly on page three > and so on and so forth. And then we turned the last page: what's there? > After a while I suggested: "nothing". I remained silent for some time .. > But then, very enthousiastic: nothing, nothing, nothing !! So before that your child would have been one of them.
From: Han de Bruijn on 2 Oct 2006 04:29
Virgil wrote: > 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. Due to the mainstream mathematics doctrine that there is a JUMP between "the" countably infinite set of naturals and all contiguous finite sets of naturals {1,2,3,4,5 , ... , n}. Resulting in n.1/n becoming n.0 . But nature does not jump (Leibniz: "Natura non facit saltus"). Han de Bruijn |