Symmetric models of ZF and GSet for large G

Symmetric models of ZF and GSet for large G

[David Roberts]

Hi,

this has been bugging me lately, and my copy of Blass and Scedrov's
AMS Memoir on order for interlibrary loan is being slow.
...

Recall that the topos of G-sets for a large group G gives us a model
of ZF via its internal logic (use a universe if desired).

Has anyone written down the precise relation to symmetric models as
defined by set theorists? Roughly speaking, these
are submodels of generic extensions that are fixed pointwise by a
family of subgroups of the automorphism group of
the generic filter (if I have the basic idea correct).

Perhaps this is in Blass-Scedrov, and I just need to be patient...

Regards,

David Roberts


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Symmetric models of ZF and GSet for large G

[Andreas Blass]

There's a logical morphism from the topos of sets to the topos of G-sets, so the internal logics of their ZF-parts (built by iterating the power set operation starting from the empty set) look the same.  As a first step toward symmetric models, you want (whether or not the group G is large) to form the topos of continuous G-sets for some group topology on G (where the sets on which G acts are given the discrete topology).  The ZF part of the resulting topos is still logically equivalent to the original universe of sets.  But, at least in the case of small G the topos may also contain objects A such that, if you iterate power set starting with A (instead of the empty set), then new phenomena, including violations of choice, can appear.  (I haven't checked what happens with large G; the problem is that A might then also be large and thus not in your topos; whether this problem actually arises may depend on the topology you put on G.)  This power-set iteration over A produces a model of the variant of ZF that allows atoms (= urelements), and the violations of choice you get this way amount to those that set-theorists get in Fraenkel-Mostoski-Specker style permutation models.

To get to ZF (without atoms) and the symmetric models you asked about, a further step is needed, namely to force over the topos of continuous G-sets to produce an A-indexed family of pure sets (i.e., sets that don't involve atoms from A).  That forcing can move the choice-violating features of A (and iterated power sets over it) into the ZF-part of the topos (without atoms).  The models you get this way are essentially the same as symmetric submodels of forcing extensions, and that's covered in considerable detail in my memoir with Andre Scedrov.  

Andreas Blass

On 10 Nov, 2011, at 9:14 PM, David Roberts wrote:

> Hi,
> 
> this has been bugging me lately, and my copy of Blass and Scedrov's
> AMS Memoir on order for interlibrary loan is being slow.
> ...
> 
> Recall that the topos of G-sets for a large group G gives us a model
> of ZF via its internal logic (use a universe if desired).
> 
> Has anyone written down the precise relation to symmetric models as
> defined by set theorists? Roughly speaking, these
> are submodels of generic extensions that are fixed pointwise by a
> family of subgroups of the automorphism group of
> the generic filter (if I have the basic idea correct).
> 
> Perhaps this is in Blass-Scedrov, and I just need to be patient...
> 
> Regards,
> 
> David Roberts



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]