An uncountable countable set
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From: Virgil on 28 Sep 2006 13:48 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. > Then you have no other choice but to add 1 each time you > proceed. That is counting. That is nonsense. > > > > > The successors are defined > > > without counting only over a very restricted domain. The domain (but not the set) of all ordinals, which is a very large domain.
From: Randy Poe on 28 Sep 2006 13:48 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
From: Virgil on 28 Sep 2006 13:53 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.
From: Virgil on 28 Sep 2006 13:57 In article <7b55f$451ba2f8$82a1e228$18740(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Theorem: 0.33333 .. = 1/3 > > Proof: 3 ( 1/10 + (1/10)^2 + (1/10)^3 + ... ) = 3 (1/(1 - 1/10) - 1) > = 1/3 : sum of geometric series > > Han de Bruijn That does not constitute a proof without an additional proof about the sum of geometric series withe ratio less than one in absolute value. A direct proof would appeal directly to the definition of the limit of an infinite series.
From: Virgil on 28 Sep 2006 14:04
In article <451ba9ed(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451b3097(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <451a8f41(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > > >>>> The question boils down to whether 0^0 is 1. > >>> 0^0 is, in any particular context, what it is defined to be. > >>> There are contexts in which it is more useful to have it mean 1 and > >>> others where it is more useful to have it mean 0. > >>> > >>> > >> But...but...but how can you reconcile those two answers??? :o > > > > As they apply in different contexts, no need to reconcile them. > >> In which contexts do you find it more convenient for it to be 0? > > > > When one wants f(x) = 0^x to be a continuous function for x >=0. > > And, in which contexts would that be desirable? When one is dealing with the collection of functions f(x) = a^x, a >= 0 instead of the collection of functions g(x) = x^a, a >= 0. The former are related to exponential functions h(x) = exp(k*x), while linear combinations of the latter, with whole number values for a, form the polynomial functions. |