Cantor Confusion
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From: Bob Kolker on 30 Oct 2006 18:06 mueckenh(a)rz.fh-augsburg.de wrote: > > > There are only countably many finite strings over a finite alphabet, > whichever may be their definition. There are countable non recursive sets. Bob Kolker
From: Virgil on 30 Oct 2006 18:07 In article <1162241829.774988.240720(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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 A list of finite sequences can be unboundedly "broad" if, say, its nth sequence is of length n. > > 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. But there can be lines, and a diagonal, larger than any finite natural, if there is no finite natural upper bound on line length. > > Regards, WM
From: MoeBlee on 30 Oct 2006 18:21 Lester Zick wrote: > My mistake, Moe. It's just that you keep snipping all these posts. I snip so that the posts are not ridiculously long. If I ever leave out context that you feel needs to be included, then you may reinstate that context in your reply. I do not snip to materially distort context, though I recognize that even judicious snipping by its very nature omits context that, in some aspect, may be considered needed. I disagree strongly with many things you say, but of course I do not begrudge your prerogative to say whatever you like about these subjects as I reply and say what I disagree with in what you've said (which is not generous of me, but rather is, to me, just a reflection of the basic premise that everyone gets to speak his or her mind)...except I ask you please not to claim I've said things that I have not said through your incorrect paraphrases of what I said. MoeBlee
From: MoeBlee on 30 Oct 2006 18:30 Lester Zick wrote: > On 30 Oct 2006 12:12:00 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >mueckenh(a)rz.fh-augsburg.de wrote: > >> Get it right: It is nonsense to talk about infinite sets if there is no > >> axiom of infinity and, therefore, no possible definition of infinity. > > > >No, YOU need to get it right. You have it completely wrong. We don't > >need the axiom of infinity to define the predicate 'is infinite'. > > But you certainly need something you ain't got besides the adjective > "infinite" to define the predicate "infinity". I never proposed considering 'infinity' as a predicate nor defining 'infinity' as a predicate or a noun. So since the other poster mentioned the impossibility of defining 'infinity', your point is well taken if it is that I should be clear that I am not responding to the poster's exact point about 'infinity' but rather that I am commenting upon the fact that we do have definitions of 'is infinite' without having to adopt the axiom of infinity. This boils down to the fact that set theory defiines 'is infinite' but there need not be any pretension on the part of set theory to define 'infinity'. What the theory NEEDS in order to do the math that it expresses is to define 'is infnite'; while it is not needed to define 'infinity'. Whatever need there is to define 'infinity' is a need that is extra to the usual mathematical purposes of devising a set theory and definitions in it. Do you see what am saying? MoeBlee
From: David R Tribble on 30 Oct 2006 18:35
mueckenh wrote: > We cannot approximate sqrt(2) arbitrarily > close. We can visualize it by the diagonal of a square and we can name > it. But we cannot approximate it better than to an epsilon of > 1/10^10^100. It woud be nice if we could, but assuming we can manage > it, only because otherwise mahematics becomes too difficult, is a bit > too simple. So I guess the natural number t = 10^(10^10^100) + 10^(10^10^100+1) can't exist, because it takes 10^(10^10^100)+1 decimal digits to represent it. (Most of the digits are 0s except for two 1s.) |