Cantor Confusion
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From: mueckenh on 30 Oct 2006 15:52 Sebastian Holzmann schrieb: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > Every member of a list of only finite sequences is a finite sequence. > > Every diagonal of such a list is a finite sequence. That is enough. > > That is wrong. Sebastian Holzmann schrieb: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > Every member of a list of only finite sequences is a finite sequence. > > Every diagonal of such a list is a finite sequence. That is enough. > > That is wrong. All entries of the list have a finite number of letters. An infinite sequence is larger than any finite sequence. The diagonal of a list cannot have more letters than the lines. According to your logic the list can have infinitely many lines. But even if that was correct it would not facilitate an infinte diagonal. The number of diagonal elements is the minimum of columns and lines. Regards, WM
From: mueckenh on 30 Oct 2006 15:53 Bob Kolker schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Correct. And this sequence has to be defined somehow. But there are > > only countably many definitions. > > That is true, if by "definition" you mean a finitary algorithm or > recipe. If this constraint does not hold then there are aleph-1 > "definitions" > Bob Kolker schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Correct. And this sequence has to be defined somehow. But there are > > only countably many definitions. > > That is true, if by "definition" you mean a finitary algorithm or > recipe. If this constraint does not hold then there are aleph-1 > "definitions" There are only countably many finite strings over a finite alphabet, whichever may be their definition. Regards, WM
From: mueckenh on 30 Oct 2006 15:55 Sebastian Holzmann schrieb: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > "dirty" is a property which without the axiom of dirt is as well > > defined as "infinite" without the axiom of infinity. > > No. A set x (which is here to denote an element of a model M of ZF-INF) > is called "finite" if x satisfies one of the following conditions: > > 0: x does not have an element > 1: x has exactly one element > 2: x has exactly two elements > and so on > > otherwise, x is called "infinite". Where do I need the axiom of infinity > to do this? Sebastian Holzmann schrieb: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > "dirty" is a property which without the axiom of dirt is as well > > defined as "infinite" without the axiom of infinity. > > No. A set x (which is here to denote an element of a model M of ZF-INF) > is called "finite" if x satisfies one of the following conditions: > > 0: x does not have an element > 1: x has exactly one element > 2: x has exactly two elements > and so on > > otherwise, x is called "infinite". Where do I need the axiom of infinity > to do this? It is impossible to prove any "otherwise". If x has not n or less elements, then it may have n+1 elements. You will never know that x has infinitely many elements because you can test only for finite numbers. It is the same as with the natural numbers. In fact you do no know that N has infinitely many elements without the axiom of infinity. Otherwise one would not need that axiom. Regards, WM
From: mueckenh on 30 Oct 2006 15:57 William Hughes schrieb: > Let our alphebet be {0,1,2}. Let our diagonal construction be > 0->1, 1->2, 2->0. Define a finite sequence as one that has only 0's > after a certain point. The set A only has sequences that have only 0's > after a certain point. > > A begins > > 000... > 1000... > 2000... > 11000... > 12000... > > The diagonal is an unending string of ones. The set A does not contain > the diagonal. > > > > > > Thus there is no contradiction. You need a diagaonal before > > > you can get a contradiction, therefore you need set B. > > > > Why should we not construct the diagonal of these sequeces (words) of > > A? > > . > We can do this but the diagonal is not a finite sequence, so it is > not a member of A. If the list consists of finite sequences, then the diaogonal is a finite sequence too. Because it cannot be broader than the list All entries of the list have a finite number of letters. An infinite sequence is larger than any finite sequence. The diagonal of a list cannot have more letters than the lines. Regards, WM
From: David Marcus on 30 Oct 2006 16:43
Han de Bruijn wrote: > David Marcus wrote: > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>mueckenh(a)rz.fh-augsburg.de wrote: > >>>stephen(a)nomail.com schrieb: > >>> > >>>>Can you describe a continuous version of the problem where each > >>>>"unit" of water has a well defined exit time? A key part of > >>>>the original problem is that the time at which each ball is > >>>>removed is defined and reached. This is crucial to the problem. It is > >>>>not just a matter of rates. If you added balls 1-10, then 2-20, > >>>>3-30, ... but you removed balls 2,4,6,8, ... then the vase is > >>>>not empty at noon, even though the rates of insertions and removals > >>>>are the same as in the original problem. So you cannot just > >>>>say the rate is 10 in and 1 out and base an answer on that. > >>> > >>>The answer for any time t *before noon* is independent of the chosen > >>>enumeration of the balls. Doesn't that fact make you think a bit > >>>deeper? > >> > >>And add this to the fact that noon and beyond cannot exist > >>in this problem. > > > > Are you still doing physics, water, and a finite number of molecules? > > Let us know when you switch to mathematics. > > I'm DOING mathematics. Mathematics is NOT independent of Physics. I guess your definition of "mathematics" is different from mine. -- David Marcus |