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). In my last email, I showed that omega_1-dependent choice is false. In fact,
On Jun 28 2011, Paul Levy wrote: there is a simple argument showing that if alpha has cofinality greater than omega then alpha-dependent choice is false. Let A_i be the set of all finite increasing sequences s of ordinals less than alpha such that only the final term of s is greater than or equal to i. Let the map A_i,j: A_i ---> A_j send a sequence s from A_i to the unique initial segment of s lying in A_j. An element of the limit of A would be a cofinal sequence for alpha of length at most omega, so if alpha has cofinality greater than omega then this limit is empty. Nathan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]