Re: Inconsistent sets
- From: Timothy Little <tim-usenet@xxxxxxxxxxxxxxxxxx>
- Date: Sun, 3 Jul 2005 23:59:58 +0000 (UTC)
mueckenh@xxxxxxxxxxxxxxxxx wrote:
> The wording was chosen in direct translation of Cantor's text: (German:
> Folge means sequence).
Terminology changes with time.
> This is, in my eyes, a misinterpretation. The axiom of infinity
> E A with { } e A and if a e A then a u {a} e A
> is nothing but a translation of Peano's
> 1 e N and if n e N then n+1 e N.
Peano's axioms aren't a set theory.
> It does not guarantee an (actually) infinite set.
It guarantees the existence of a set for which a bijection exists
between the set itself and a strict subset, so yes, it does guarantee
an actually infinite set.
> But even given the case, the axiom would guarantee an infinite set,
> then it was in contradiction with itself, because
> 1) the ordinal of this set, omega, cannot be an element of N
Correct.
> 2) the axiom creates or guarantees only natural numbers (never omega).
The natural numbers don't need the axiom of infinity. Every natural
number is constructible without it. The axiom of infinity says that
there is a set (N) that contains them all. The subset well-ordering
of N then defines the order-type omega. Omega is not a set, I suspect
where you're talking about omega as a set you really mean some set
that is order isomorphic to N.
> No. Omega is not. I don't see it in von Neumann's construction of
> ordinals, nor in Zermelo's.
Omega is not a set. It's the order type of N.
> What does ZFC help us? There is no application outside mathematics and
> there is no meaningful application inside, oher than confusing its
> disciples by making them erroneously believe that mathematics had a
> solid foundation.
Oh. I hadn't considered that you might say that. *Plonk*
- Tim
.
- Prev by Date: Re: Cantor and the binary tree
- Next by Date: Re: Cantor and the binary tree
- Previous by thread: Re: Inconsistent sets
- Next by thread: Re: Inconsistent sets
- Index(es):
Relevant Pages
|
