Here's a counterexample: for alpha < omega_1, let F(alpha) be the set of injections f: alpha --> omega for which the complement of the image of f is infinite, and for alpha < beta let F(beta) --> F(alpha) be defined by restriction. Peter Johnstone On Tue, 28 Jun 2011, Paul Levy wrote:
Dear all,
Let alpha be an ordinal. Let $alpha be the totally ordered set of ordinals below alpha.
"Alpha-dependent choice" is the following statement:
for any functor A : $alpha ^ op ---> Set, if A_i is nonempty for all i < alpha, and A_i,j : A_j ---> A_i is surjective for all i <= j < alpha, then the limit of A is nonempty.
If alpha has a cofinal omega-sequence (i.e. an omega-sequence of ordinals < alpha whose supremum is alpha), then alpha-dependent choice follows from dependent choice.
I would think that, if alpha doesn't have a cofinal omega-sequence, then alpha-dependent choice is false. Is there a known counterexample? E.g. in the case alpha = omega_1 (the least uncountable ordinal).
Thanks, Paul
-- Paul Blain Levy School of Computer Science, University of Birmingham +44 (0)121 414 4792 http://www.cs.bham.ac.uk/~pbl
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