[FOM] Global Choice

Thomas Forster T.Forster at dpmms.cam.ac.uk
Wed Apr 7 04:53:07 EDT 2010

Is this an example of a general phenomenon, in ZF (possibly even without 
choice or foundation)?  Consider a definable equivalence relation (as it 
might be: equipollence).  Suppose we make the assumption that for any 
subset $x$ of $V$ there is $f:x \to V$ such that $(\forall u,v \in x)(f(u) 
= f(v)$ iff $u$ and $v$ are equivalent$)$.  (This is a *set* theory). 
Now consider the *class* theory that is GB (again, possibly without choice 
or foundation) plus an axiom that says there is a global class function 
$f$ such that $(\forall u,v)(f(u) = f(v)$ iff $u$ and $v$ are 
equivalent$)$.  Is this class theory a conservative extension of the 
coresponding set theory? Or do we need choice & foundation for the forcing 
argument to work?

On Mon, 
5 Apr 2010, Robert Solovay wrote:

> Thomas,
> The answer is yes. The result is due to a lot of people (including
> Jensen and myself). Basically one forces to add a generic
> well-ordering of the universe without adding new sets. Felgner
> published this in 1971:
> F. published this. (Fund, Math, 71(1971), pp. 43--62)
> I  also  found the following relevant paper of Gaifman:
> Global and local choice functions
> Journal	Israel Journal of Mathematics
> Issue	Volume 22, Numbers 3-4 / December, 1975
> Pages	257-265
> Global and local choice functions
> Abstract  We prove, by an elementary reflection method, without the
> use of forcing, that ZFGC (ZF with a global choice function) is a
> conservative extension of ZFC and that every model of ZFC whose
> ordinals are cofinal (from the outside) with? can be expanded to a
> model of ZFGC (without adding new members). The results are then
> generalized to various weaker forms of the axiom of choice which have
> global versions.
> --Bob Solovay
> On Mon, Apr 5, 2010 at 1:42 AM,  <T.Forster at dpmms.cam.ac.uk> wrote:
>> Is Goedel-Bernays + global choice a conservative extension of ZFC..?
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