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

