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Topic: Another AC anomaly?
Replies: 43   Last Post: Dec 21, 2009 8:08 AM

 Messages: [ Previous | Next ]
 K_h Posts: 419 Registered: 4/12/07
Re: Another AC anomaly?
Posted: Dec 18, 2009 7:56 PM

"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
news:KuuL76.5GA@cwi.nl...
> In article <eaCdnfgmKeFaorbWnZ2dnUVZ_hmdnZ2d@giganews.com>
> "K_h" <KHolmes@SX729.com> writes:
> ...

> > > Under the standard
> > > construction each natural is `accumulated' in the
> > > limit
> > > set, since each set persists beyond its introduction,
> > > and
> > > this preserves the notion of n growing bigger as one
> > > proceeds: |n| is 0,1,2,3,... and is N in the limiting
> > > case.

> >
> > Yo, it should be |N| in the limiting case!

>
> Eh? With your definition, lim {n} = {N} (as you stated
> earlier) and so
> lim n = N. But now you actually state (as |n| = n), lim n
> = |N|. Are
> N and |N| equal or not?

Yes, in this case they are equal.

> > > proceeds: |n| is 0,1,2,3,... and is N in the limiting
> > > case.

"N in the limiting case" was a typo, I meant "|N| in the
limiting case" but it is correct in either case. This post
had nothing to do with my definition. Using just
wikipedia's definitions for a limit set and the standard
construction of the naturals:

lim (n-->oo) n = N //for sets.
|lim (n-->oo) n| = lim (n-->oo) |n| = |N| //for cardinals

Using the standard construction of the naturals, it is a
theorem in ZF that if n is a finite ordinal then |n|=n.
There is no contradiction here because aleph_0 is defined to
be the first ordinal w=N so the limit ordinal w equals
lim(n-->oo) n = N and since |N|=aleph_0=w=N, |N|=N in this
case.

k

Date Subject Author
12/12/09 Jesse F. Hughes
12/13/09 K_h
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/16/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/21/09 Dik T. Winter
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/17/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/15/09 K_h
12/16/09 Jesse F. Hughes
12/17/09 Dik T. Winter
12/17/09 Jesse F. Hughes
12/16/09 Dik T. Winter
12/15/09 ross.finlayson@gmail.com
12/13/09 K_h
12/13/09 Jesse F. Hughes
12/13/09 Jesse F. Hughes
12/14/09 Ilmari Karonen
12/14/09 Jesse F. Hughes
12/15/09 Chas Brown