
Re: Another AC anomaly?
Posted:
Dec 16, 2009 7:44 AM


In article <hNdneOj6K8oz7XWnZ2dnUVZ_rKdnZ2d@giganews.com> "K_h" <KHolmes@SX729.com> writes: > "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message > news:Kuq5DH.18H@cwi.nl... ... > > > 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}={}?
That is not important for me. With *your* definition and the construction of the natural numbers with n+1 = {n}, we *get* that lim(n > oo) {n} = {} (note: set limit).
> 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.
We are talking about lim(n > oo) {n} which is the limit of a sequence of sets, and not about lim(n > oo) n which may or may not be the limit of a sequence of sets, depending on the actual construction of the natural numbers.
But if I understand you well, your opinion is not that lim(n > oo) {n} can be {N} or nonexisting, depending on the way the natural numbers are constructed?  dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

