Cantor Confusion
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From: MoeBlee on 30 Oct 2006 15:18 mueckenh(a)rz.fh-augsburg.de wrote: > Every member of a list of only finite sequences is a finite sequence. Correct. > Every diagonal of such a list is a finite sequence. Incorrect since there are infinite sequences (lists) of finite sequences. A list is a sequence. Some sequences are finite and some sequences are denumerable (and there can be sequences on uncountable ordinals too). Even if every member of the range of the sequence S is a finite sequence, it still may be that S is itself a denumerable sequence (it is a denumerable list of finite lists), and S may very well have a diagonal that is a denumerable sequence. MoeBlee
From: MoeBlee on 30 Oct 2006 15:25 mueckenh(a)rz.fh-augsburg.de wrote: > What kind of existence do you have in mind? These things do not exist > unless they exist in some mind. But you say that they do not exist in > any mind. So what *is* existence in this case? That's a philosophical question for each one to answer for him or herself. However, at the very least, by 'something exists with the property described by formula phi', one may simply mean that the sentence Ex phi(x) is a theorem of the theory. Moreover, someone such as Abraham Robinson (founder of non-standard analysis) denies that he can conceive of an infinite set and even denies the meaningfulness of the claim that infinite sets exits, except that he recognized the usefullness of infinite sets as ideal entities in mathematical theories. MoeBlee
From: MoeBlee on 30 Oct 2006 15:28 Lester Zick wrote: > You mean the question where you demand I research and justify your > opinions for you, Moe? I didn't demand anything of you in a question, let alone that I've never demanded of you that you justify my opinions. MoeBlee
From: MoeBlee on 30 Oct 2006 15:30 Sebastian Holzmann wrote: > I just wanted to add some additional point. You are, of course, right in > your argument. And your point is well taken too, as an additional layer of understanding of the matter. MoeBlee
From: MoeBlee on 30 Oct 2006 15:39
mueckenh(a)rz.fh-augsburg.de wrote: > "Given any model of ZF-INF, one cannot be sure if there are infinite > sets" is as silly an argument as talking about sets of rats or > rabbits, or sets being clean or dirty. No, it means that simply dropping the axiom of infinity does not permit us to conclude that there do not exist infinite sets. In other words, simply dropping the axiom of infinity from ZFC does not permit us to make arguments in ZFC-I with an ongoing premise that all sets are finite. If you want to make arguments in ZFC-I with the premise that all sets are finite, then you must adopt an axiom that declares that all sets are finite; just dropping the axiom of infinity is not sufficient. MoeBlee |