[FOM] Global Choice

[Ali Enayat]

In his recent posting (April 5, 2010), Thomas Forster asks:

>Is Goedel-Bernays + global choice a conservative extension of ZFC?

The answer is a resounding "yes".

The usual way of seeing this is to show, via forcing, that every
countable (but not necessarily well-founded) model M of ZFC can be
expanded to a model of GB + global choice. The conservativity result
then follows by the completeness theorem of first order logic.

The forcing conditions are (local) choice functions, ordered by
inclusion. The generic object F is easily seen to be global choice
function over M., but some work is required to verify that (M,F)
satisfies the replacement scheme in the extended language {epsilon,
F}. Then if C is the collection of parametrically definable subsets of
(M,F), then (M,C) is a model of GBC + global choice.

I recall reading somewhere many people independently discovered the
above result (including Cohen, Solovay, Kripke, and Felgner).
Felgner's account can be found in the following reference.

U. Felgner, Choice functions on sets and classes. Sets and classes (on
the work by Paul Bernays), pp. 217--255. Studies in Logic and the
Foundations of Math., Vol. 84, North-Holland, Amsterdam, 1976.

Also note that Gaifman found a new proof of the above conservativity
result without forcing, see below:

H. Gaifman, Global and local choice functions. Israel J. Math. 22
(1975), no. 3-4, 257--265.

Best regards,

Ali Enayat