[FOM] The Minimal Model of ZF
Ilya Tsindlekht
eilya497 at 013.net
Thu Jan 3 05:01:05 EST 2008
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.
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