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From: N.Bowler@dpmms.cam.ac.uk
To: Paul Levy <pbl@cs.bham.ac.uk>
Cc: categories list <categories@mta.ca>
Subject: Re: ordinal dependent choice
Date: 29 Jun 2011 09:32:15 +0100	[thread overview]
Message-ID: <E1QbukV-0002fY-IG@mlist.mta.ca> (raw)
In-Reply-To: <E1QbemH-0007q1-45@mlist.mta.ca>

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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  reply	other threads:[~2011-06-29  8:32 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 [this message]
2011-06-29  9:12 ` Prof. Peter Johnstone
2011-06-30 16:07 ` Paul Levy

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