ordinal dependent choice

ordinal dependent choice

[Paul Levy]

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/ ]

Re: ordinal dependent choice

[N.Bowler]

On Jun 28 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).
In my last email, I showed that omega_1-dependent choice is false. In fact,
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


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Re: ordinal dependent choice

[Prof. Peter Johnstone]

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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Re: ordinal dependent choice

[Paul Levy]

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/ ]