K_h
Posts:
419
Registered:
4/12/07


Re: Another AC anomaly?
Posted:
Dec 18, 2009 3:39 AM


"K_h" <KHolmes@SX729.com> wrote in message news:SqSdnSUF1q8FobbWnZ2dnUVZ_uCdnZ2d@giganews.com... > > "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.
Yo, it should be N in the limiting case!
k

