[FOM] Global choice and ZF

Allen Hazen allenph at unimelb.edu.au
Sat Apr 4 02:59:32 EDT 2009

I've never read it, but you might find relevant information in Ulrich
Felgner, "Choice functions on sets and classes" (pp. 217-255 in Gert Müller,
ed., "Sets and Classes," Amsterdam: North-Holland, 1976: this is the yellow
"Studies in Logic" volume that contains the reprinting of Bernays's JSL
articles).  Corollary 4.1 (p. 243) says a sentence in the language of ZF is
provable in ZF + (local) Choice iff it is provable in NB + (global) Choice.

It is assumed in this that ZF and NB include the Axiom of Foundation:
apparently things are -- or at least in 1976 were -- a bit dicier in the
absence of foundation.  (We don't know anyone interested in systems without
foundation, though, do we?)

I really must get around to reading this some day: I thought I had a REALLY
easy proof of the result, but if it merited an article of this length I must
have overlooked some essential complication!
Allen Hazen
Philosophy (PASI)
University of Melbourne

On 3/4/09 7:01 PM, "T.Forster at dpmms.cam.ac.uk" <T.Forster at dpmms.cam.ac.uk>

> The feedback I am getting suggests that the truth of the matter is that
> NGB + global choice is an extension of NGB + set choice that is
> conservative for the language of pure sets.
>  Can anyone confirm this, preferably with a reference?

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