[FOM] The Minimal Model of ZF

[Steven Gubkin]

From: joeshipman at aol.com
> On the other hand, if M is a set, then internally M satisfies V=M, 
> and there IS a first set in M which, externally, we can name, but 
> which M does not "know" has a name.


You state that "for every element x in M there is a formula A(y) 
in the language of set theory such that x is the unique element of M 
satisfying A. Thus in M every element can be 'named'". Since there 
are countably many formulas in the language of set theory, we know 
that M is 
countable "externally". When you talk about M "knowing" or "not 
knowing" that it has a given element of M has a name, what do you mean 
exactly? As far as set theory goes, all of the formulas are "about" 
sets; none of them are "about" formulas, unless you are representing 
the language of set theory within M (say by Godel numbering, which is 
possible since w must be in M). Then there will be an external 
bijection between {internal reprentations of formulas} with {members 
of M}, but there could be no M-function making this assignment 
(otherwize M would be a set internally). So it looks like the fact 
that M fails to contain such a function internally resolves the 
paradox. Am I missing something? It seems like there was more to your 
argument, but I can't reason through it without understanding this bit 
first.

p.s. I have not written about set theory very often, and I found it 
very difficult to express what I was trying to say. If anyone has any 
questions, or suggestions about how to better phrase things, it would 
help me alot.