FOM: The logical, the set-theoretical, and the mathematical: Reply to Ketland

[Kanovei]

> Date: Wed, 13 Sep 2000 08:46:32 +0100
> From: "V. Yu. Shavrukov" <vys1 at mcs.le.ac.uk>
 
> answering the following questions:
> 
> What is the result of allowing free variable epsilon terms into various
> set existence schemas?  Is it always, or at least "in most cases" equivalent to
> adding AC?
> 

ZFC + the epsilon-term seems to be the same as ZFC + Global Choice, 
or ZFGC. 
It is known (they refer to Felgner, 1971) that ZFGC is 
a conservative extension of ZFC, that is, any theorem of 
ZFGC in pure membership-language is a theorem of ZFC. 

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. 

Vladimir Kanovei