[FOM] A confession

Colin McLarty colin.mclarty at case.edu
Fri Sep 6 09:23:10 EDT 2013

```Hi,

Yes.  It became an absorbing problem for me but I could not add anything. I

Various kind people have pointed out that your proof that the world of NF
> sets is not cartesian-closed puts severe difficulties in the path of any
> such project. It seems that all you need is that NF has a universal set.
>
> **Does this mean that the same result holds for positive set theory?**
>

No, this proof as written does not give that.  The pivot point in my
argument is to get apart from details about which definition of ordered
pairs to use and down to the elementary proof that no topos can have an
object to which each object has some monic arrow.  The NF axioms do imply
each part of that except but cartesian closedness, so they cannot also have
cartesian closedness.

Without being sure what all the options are for defining 'function' in
positive set theory it seems clear the natural definitions will not imply
subsets have characteristic functions.  The subobject classifier axiom of a
topos will fail, and so this argument will not show any other axiom has to
fail (of course others may have to fail, for other reasons).  Part of this
I can prove and part I hope to make plausible.

It is provably impossible in positive set theory that every subset S of a
set A has a characteristic function from A to a two element set.  Such a
characteristic function would imply subset S has a (boolean) complement in
A.  And I hope it is plausible that if the sets were to form a topos it
would be well pointed and so it would have a two-element truth value
object, but for lack of a classification of all the ways to define function
in positive set theory I cannot make this a proof.

Thanks for a fun question.  I wish i had more definite answers.

best, Colin

I think my proof is useful all the same, since it works for KF as well
> (indeed for any set theory in which the singleton function is not reliably
> a set, locally)
> Elaine Landry has kindly supplied me with a pdf of your paper (tho' i now
> realise i had a copy on my laptop all the time!!!) So i can get stuck in.
>
>   I would glad to hear any thoughts you have on this matter..
>
>       v best wishes
>
>           tf
>
>
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