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K_h
Posts:
419
Registered:
4/12/07


Re: Another AC anomaly?
Posted:
Dec 15, 2009 10:23 PM


"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message news:Kuq5DH.18H@cwi.nl... > In article <Kup2uH.JC7@cwi.nl> "Dik T. Winter" > <Dik.Winter@cwi.nl> writes: > > In article > > <r_6dnUXDv4pMfbvWnZ2dnUVZ_tWdnZ2d@giganews.com> "K_h" > > <KHolmes@SX729.com> writes: > > > "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message > > > news:KunF0v.9y9@cwi.nl... > > ... > > > > By what definition is: > > > > lim(n > oo) n = N? > > > > > > Any of the wikipedia definitions. Here is the proof > > again. > > ... > > > Theorem: > > > lim(n >oo) n = N. Consider the naturals: > > > > > > S_0 = 0 = {} > > > S_1 = 1 = {0} > > ... > > This presupposes a particular construction for the > > natural number. There are > > other constructions that are consistent with ZF. Is the > > limit valid for all > > those possible models? > > For starters, try it with > 0 = {} > n+1 = {n} > which is a valid construction of the naturals in ZF. > > Even with your definition > lim sup(n > oo) {n} = {}
Why is it so important to you to have a limit definition and a construction of the naturals such that lim(n>oo){n}={}? The general idea of a limit is that the limiting state is what you get when you go through all sequences. If one defines the naturals as you have done above then the general notion of a limit suggests that the limiting state should be something like:
{...{{{{{{...{}...}}}}}}...} = limit
We could construct a defintion of a limit so that this is the end result but it may be that a better definition for the limiting case of 0={} and n+1={n}is a defintion where lim(n > oo)n does not exist.
k



