Re: size_question_encore

[Michael Shulman <mshulman@ucsd.edu>]

On Wed, Jul 13, 2011 at 9:10 PM, Toby Bartels
<categories@tobybartels.name> wrote:
>>the axiom of collection, which implies that we can find some *set* of
>>objects containing *at least one* limit for every finite diagram in
>>the original small subcategory; and then we can iterate countably many
>>times to obtain a small category which contains at least one limit for
>>any finite diagram therein.
>
> The axiom of collection guarantees only *some* appropriate set of objects,
> so you need to choose one.  To iterate this countably many times,
> you might need dependent choice.

That's a good point.  However, I think we can get around it as
follows.  We can make finitely many choices without any axiom of
choice.  Thus, for any natural number n, by applying collection n
times, we can find *some* n^th iterate of the "construction".
(Formally, we prove this by induction on n.)  Applying the axiom of
collection again over the natural numbers, we obtain a set which
contains at least one n^th iterate of the "construction" for every
natural number n.  Taking the union of this set, we should obtain a
set of objects whose corresponding full subcategory contains at least
one limit of every finite diagram therein.

Mike


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