Re: size_question

[James Lipton <jlipton@wesleyan.edu>]

Eduardo:
   For a small cat of finite sets: Why not use the Von Neumann hierarchy (up
to omega) for objects, and all set
functions as arrows. These are the "hereditarily finite sets"

    V(0) = empty set

   V(n+1) = P(V(n))

V(omega) = U{V(n): n in omega}

I would not call it "the" cat of finite sets, since there uncountably many
countable models of ZF. Obviously the choice of ZF, rather than say Zermelo
set theory or some other foundation is also pretty arbitrary.
Best,
   J. Lipton

On Tue, Jun 28, 2011 at 1:39 PM, Eduardo Dubuc <edubuc@dm.uba.ar> wrote:

> This is a naive question on non naive foundations.
>
> Consider the inclusion S_f C S  of finite sets in sets.
>
> Is the category S_f closed under finite limits and at the same time small ?
>
> For example, there are a proper class of singletons, all finite. Thus a
> proper class of empty limits.
>
> Question, which is the small category of finite sets ?, which are its
> objects ?.
>
> A small site with finite limits for a topos would not be closed under
> finite limits ?
>
> etc etc
>
> But, more basic is the question above: How do you define the small
> category of finite sets ?
>
> Or only there are many small categories of finite sets ?
>
> You can not define a finite limit as being any universal cone because
> then you get a large category.
>
> Then how do you determine a small category with finite limits without
> choosing (vade retro !!) some of them. And if you choose, which ones ?
>
> The esqueleton is small but a different question !!
>
> e.d.
>

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