[FOM] Global Choice

[Ali Enayat]

In his recent posting (April 7), Thomas Forster has asked whether the
forcing argument establishing the conservativity of GBC over ZFC can
be generalized as follows (I have edited out the LaTeX commands).

>Consider a definable equivalence relation (as it might be: equipollence).  Suppose we make the assumption that for any
>subset x of V there is a function f with domain x such that for all u and 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 for all u,v, f(u) = f(v) iff u and v are equivalent.

>Is this class theory a conservative extension of the corresponding set theory? Or do we need choice & foundation for the forcing
argument to work?

The forcing argument definitely does not need choice.

For example, one can use forcing to show that if M is a countable
model of ZF that satisfies the Kinna-Wagner selection principle KW
(which asserts that every set can be injected into the power set of
some ordinal), then M can be expanded to (M,F), where F is a global
Kinna Wagner function (injecting the universe into the class of
subsets of ordinals), and (M,F) satisfies ZF in the extended language
{epsilon, F}. Of course ZFC proves KW, but it is known that there are
models of ZF+KW in which AC fails.

As for foundation: I am *fairly certain*  that it is not essential for
the forcing proof, but I would have to re-examine the proof to be
sure.

The key axiom that makes the forcing proof go through is the
replacement scheme, which allows us to reflect any global first order
property of the universe onto a local piece of the universe (even in
the absence of the foundation axiom).

Let me close by re-iterating a related question that I posed several
years ago in FOM:

Question: is Zermelo set theory with global choice conservative over
Zermelo set theory with local choice?

Regards,

Ali Enayat