K_h
Posts:
419
Registered:
4/12/07


Re: Another AC anomaly?
Posted:
Dec 17, 2009 1:20 AM


"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message news:KuqwqJ.24o@cwi.nl... > 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... > ... > > 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.
For the sets n, lim(n>oo)n is the limit of a sequence of those sets. What lim(n>oo){n}={} really means is that each {n} does not persist after it first appears. The essence of the defined limsup and liminf is that they only contain those sets that persist, as members of subsequent sets, after they first appear. For the nonnaturals, {n}, lim(n>oo){n}={} just says nothing is accumulated. For the naturals, n, lim(n>oo)n=N just says that everything is accumulated. There are many ways a limit of a sequence of sets can be defined. Wikipedia gives two such examples but these are not the only two options. Using the above notion of accumulation, lim(n>oo){A_n} can be defined by a set containing the accumulation of A_n. For example, this definition says lim(n>oo){n}={N} and its sequence and limit look like.
{{}}, {{0}}, {{0,1}}, {{0,1,2}}, ... > {{0,1,2,3,,...}}
So this definition is motivated by the intuitive idea of what a limit is in cases like these. Like many definitions, this one has disadvantages: some theorems true in other definitions will not be true with this one.
> > > > 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? > > > > > Why do you ask? There are many ways a limit can be > > defined > > in ZF but the definition should embody the general idea > > of > > what a limit is. > > But the definition ought to be such that the limit of a > sequence does not > depend on the exact construction of the sequence. That is > that > lim(n > oo) {n} > should be independent on the way the natural numbers are > constructed.
lim(n >oo) n = N is true for the standard definition of natural numbers using just the wikipedia definitions for the limit of a sequence of sets.
k

