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From: "Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>
To: Paul Levy <pbl@cs.bham.ac.uk>
Cc: categories list <categories@mta.ca>
Subject: Re: ordinal dependent choice
Date: Wed, 29 Jun 2011 10:12:58 +0100 (BST)	[thread overview]
Message-ID: <E1Qbulj-0002hQ-7n@mlist.mta.ca> (raw)
In-Reply-To: <E1QbemH-0007q1-45@mlist.mta.ca>

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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  parent reply	other threads:[~2011-06-29  9:12 UTC|newest]

Thread overview: 4+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2011-06-28 13:12 Paul Levy
2011-06-29  8:32 ` N.Bowler
2011-06-29  9:12 ` Prof. Peter Johnstone [this message]
2011-06-30 16:07 ` Paul Levy

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