Re: abundance of irrationals!)
- From: Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx>
- Date: Tue, 03 May 2005 12:21:07 -0600
In article <1115138549.844395.225970@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:
> Randy Poe wrote:
>
> > > But how do you find Card(N) = aleph_0 > n e N from the
> Peano-axioms?
> >
> > I could be wrong, but I think aleph_0 is defined
> > to be card(N).
> >
> > What you can easily prove from the Peano axioms is
> > that card(N) is larger than any finite value.
>
> But there is no proof that this Card(N) is a meaningful notion at all.
The definition of 'cardinality' is an equivalence class under the
equivalence relation of being in one to one correspondence (having a
bijection from one to the other).
Thus any set has a well defined cardinality.
The only issue is whether there is an total order or only a partial
order on (infinite) cardinalities under the definition that
Card(A) <= Card(B) iff there is an injection from A to B.
.
- Follow-Ups:
- Re: abundance of irrationals!)
- From: mueckenh
- Re: abundance of irrationals!)
- References:
- Re: abundance of irrationals!)
- From: W. Mueckenheim
- Re: abundance of irrationals!)
- From: Randy Poe
- Re: abundance of irrationals!)
- From: mueckenh
- Re: abundance of irrationals!)
- From: Randy Poe
- Re: abundance of irrationals!)
- From: mueckenh
- Re: abundance of irrationals!)
- Prev by Date: Re: abundance of irrationals!)
- Next by Date: Re: Teaching Myself Mathematics
- Previous by thread: Re: abundance of irrationals!)
- Next by thread: Re: abundance of irrationals!)
- Index(es):
Relevant Pages
|
Loading
