
Re: Another AC anomaly?
Posted:
Dec 14, 2009 4:27 PM


["FollowupTo:" header set to sci.math.] On 20091214, Jesse F. Hughes <jesse@phiwumbda.org> wrote: > "Jesse F. Hughes" <jesse@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.
 Ilmari Karonen To reply by email, please replace ".invalid" with ".net" in address.

