# FOM: epsilon terms and choice

V. Yu. Shavrukov vys1 at mcs.le.ac.uk
Sun Sep 24 18:59:52 EDT 2000

Dear Professor Kanovei,

>> I am not quite sure what you mean by ZFC + Global Choice'
>> in the first sentence.  Is it  GB + GC, or
>> ZFC + schema: definable classes exist + GC
>> (+ extensionality for classes),  or something else?
>
>ZFGC is a theory in the language including =, \in, and G,
>an atomic functional symbol,
>the axioms of ZFGC include all of ZFC (G is allowed to occur
>in the schemata) and the axiom saying that if x is a non-0 set
>then G(x) is defined and belongs to x.
>
>V.Kanovei

Thank you for your explanations.  I see what you mean.

Still, to model epsilon terms with  G,  I seem to need
Foundation:  G  chooses from all sets, whereas  epsilon
terms choose from possibly class-sized collections, so
under Foundation you just restrict to elements of minimal
rank and get sets.  Is there a way to do it differently?

It's not that I have anything against Foundation.
Rather, I seem to remember somebody commenting that
epsilon terms are easily explained away in terms of
less exotic things in a fairly general setup.
The explaining away that depends on Foundation does
not appear too general to me.

Also, you write:

>As far as I know it is still open whether ZGC
>(Zermelo + Global Choice) is a conserv. ext. of ZC
>(Zermelo with Choice) in the same sense,
>it seems the class-forcing construction of
>Felgner needs Replacement in the ground model
>even to prove Separation in the extension.

On the other hand, Mathias, in his recent paper
(Bulletin London M.S., this year, number 5) tacitly
equates "Bourbachisme" with  ZC.

I am not however sure that this should be interpreted
as him saying that  Z+epsilon  is conservative over
ZC,  but this would be my natural reaction.
Any comment?

sincerely,
V.Shavrukov

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