Cantor Confusion
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From: MoeBlee on 30 Oct 2006 17:38 mueckenh(a)rz.fh-augsburg.de wrote: > 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. 0 1 2 3 4 5 6 7 8 9 ................ .................. .................... infinitely downward for an infinite list of finite lists. The diagonal is 0 2 5 9 14 ... infinitely across. All entries in the infinite list are finite lists. The infinite list is longer than any finite list. The diagonal of the list is infinite. And that be formalized easily in set theory. MoeBlee
From: Virgil on 30 Oct 2006 17:39 In article <1162241530.355257.54540(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. It can be, and must be, longer than any single line, and since there is no finite upper bound on line length, there can be no finite upper bound on diagonal length either, which means the diagonal can be infinite. > > According to your logic the list can have infinitely many lines. But > even if that was correct it would not facilitate an infinte diagonal. Sure it would. If line n is of length >= n, and the diagonal differs from it in place n, for each of infinitely many values of n, then the diagonal is infinite. > > The number of diagonal elements is the minimum of columns and lines. Actually, it would be at least the maximum of both, and if neither is finitely bounded, then the diagonal must be infinite. > > Regards, WM
From: Virgil on 30 Oct 2006 17:41 In article <1162241587.443246.227770(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. But without both finitenesses, countability is not guaranteeable.
From: Sebastian Holzmann on 30 Oct 2006 17:41 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > Sebastian Holzmann schrieb: >> mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: >> 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. 1. Any given set either satisfies one of the conditions, or it doesn't any. If you have objections to that, you leave the paths of standard logic, and your reasoning bears no relevance to standard mathematics. 2. The axiom of infinity is added to ensure that a set "like N" exists in _every_ model of the theory. If it isn't added, there might be models that do not have an "N" and some who do. Learn about logic, model theory and set theory!
From: Virgil on 30 Oct 2006 18:03
In article <1162241707.218096.63760(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. But unless you know that x has some particular finite number of members, or no more than some particular finite number of members, you cannot be sure it is not infinite either. > > Regards, WM |