[FOM] The Minimal Model of ZF

joeshipman@aol.com joeshipman at aol.com
Thu Jan 3 19:33:12 EST 2008

>-----Original Message-----
>From: Ilya Tsindlekht <eilya497 at 013.net>
>> extending the axiom system "ZFC + V=M" with the scheme "Anything 
>> '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

No it isn't inconsistent, you didn't read this as carefully as I worded 

I spoke about what "ZFC + M exists as a set" proves *is true in M*, not 
what it proves is true in V. "ZFC + M exists as a set" proves that the 
following are true in M:

1) ZFC
2) V=L
3) M does not exist as a set
4) Con(ZFC)

Taking "ZFC +V=M" as an axiom system allows us to get 1), 2), and 3), 
but not 4). Extending it with my further scheme "assume everything that 
would be true in M if M were a set" gives us 4), so it is a genuine 
extension, and also a consistent extension, since each axiom in that 
axiom scheme is true in M. ("M exists as a set" is NOT one of the 
sentences in the axiom scheme, it is a just a tool that allows us to 
recognize such sentences.)

-- JS
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