K_h
Posts:
419
Registered:
4/12/07


Re: Another AC anomaly?
Posted:
Dec 18, 2009 3:26 AM


"Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message news:Kusoty.176@cwi.nl... > In article <yrydnX_FwfczULTWnZ2dnUVZ_vmdnZ2d@giganews.com> > "K_h" <KHolmes@SX729.com> writes: > ... > > 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. > > But not with Zermelo's definition of the natural numbers. > Nor when we take > the natural numbers as embedded in the rational numbers.
Yes, and that was never denied. I just point out that any limit definition for a sequence of sets will give answers that violate the intuitive notion of a limit for certain constructions. There are two ways one can deal with that. First, for a given class of constructions, have another definition that does not violate the limit notion. The second way is to look at the meaning that the wikipedia definitions have for the constructions. I now think that the latter approach is superior to the first since the first caused so much misunderstanding. With the second option we can restrict ourselves just to the wikipedia definitions and define the sets n by:
n = 0 = {} n = 1 = {{}} n = 2 = {{{}}} ...
Wikipedia gives liminf(n>oo) n = 0. What this means is that nothing is `accumulated' in a limit set since each set does not persist past its introduction. So the notion of n growing bigger, as one tends to the limit, is not embodied here since n is 0,1,1,1,1... as one proceeds and is 0 in the limiting case. 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.
k

