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/ ]

Thanks to all the people who sent me counterexamples, references and Aronszajn trees. I found the following document helpful: http://math.berkeley.edu/~gbergman/papers/unpub/emptylim.pdf It seems that Higman and Stone's result mentioned there is the same as Aronszajn's. Paul PS Of course I should have stated that alpha is a positive limit ordinal, for otherwise alpha-dependent choice holds trivially. On 28 Jun 2011, at 14:12, 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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

-- Paul Blain Levy School of Computer Science, University of Birmingham +44 (0)121 414 4792 http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]