Re: ordinal dependent choice

[Paul Levy <pbl@cs.bham.ac.uk>]

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
>
>
>
>
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> [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











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