[FOM] The Minimal Model of ZF

[Ilya Tsindlekht]

On Wed, Jan 02, 2008 at 01:54:27AM -0500, joeshipman at aol.com wrote:
> To sharpen the "paradox" I refer to at the end of my previous post:
> 
> If ZF has a standard model, then M, the minimal model, exists as a set, 
> and Th(M) is a well-defined set of sentences that includes the axioms 
> of ZFC, "V=M", and lots of other things. If M does not exist as a set, 
> then Th(M) is no more definable than Th(V); but what goes wrong with 
> the previous definition, exactly?
Exactly this: M does not exist.
> 
> Another way to put it: assuming M exists as a set, it is a unique and 
> well-defined set, but is there any sentence that we can prove M 
> satisfies that is not already a logical consequence of "ZFC+V=M"? 
> Assuming Con(ZF), of course Con(ZF) is such a sentence, because M 
> satisfies the true sentences of arithmetic which are absolute and if M 
> exists as a set then ZF is consistent; but I'd like to see how much 
> further this can be taken.  ("Taking it further" corresponds to 
> extending the axiom system "ZFC + V=M" with the scheme "Anything which 
> 'ZFC + M exists as a set' proves is true in M is true in V".)
> 
Since V=M implies 'M does not exist (as a set)' such extension is
inconsistent.