
Re: Another AC anomaly?
Posted:
Dec 15, 2009 1:35 AM


On Dec 14, 5:28 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > Ilmari Karonen <usen...@vyznev.invalid> writes: > > ["FollowupTo:" header set to sci.math.] > > On 20091214, Jesse F. Hughes <je...@phiwumbda.org> wrote: > >> "Jesse F. Hughes" <je...@phiwumbda.org> writes: > > >>> But the standard topology on N is the discrete topology, too! Thus, > >>> the standard definition of sequence convergence on N is inherited via > >>> the subspace topology from Set. That is, a sequence > >>> {a_n  n in N} c N converges (in N) to m iff > > >>> (E k)(A j > k) a_j = m. > > >>> This is (unless I'm just buttwrong) the same as the definition of > >>> sequence convergence on Set restricted to the subspace N. > > >> Yeah, well, I am just buttwrong, ain't I? > > > Well, not really. That's not the same as the definition of general > > set convergence, but I do believe the two definitions are equivalent > > for sequences of natural numbers, at least under any of the usual > > settheoretic constructions of the naturals. > > > In particular, under the standard construction of the naturals, where > > 0 = {} and n+1 = n union {n}, I believe the two definitions of lim sup > > and lim inf also match: this is due to the fact that, for the natural > > numbers m and n under this construction, m is a subset of n if and > > only if m <= n. > > Oh. Okay, but if I'm right, it was only coincidence. So, perhaps I > was buttright. >
I'd say you were right, but...
Cheers  Chas

