# [FOM] The Minimal Model of ZF

Steven Gubkin steven.gubkin at case.edu
Tue Jan 22 21:52:13 EST 2008

```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.

```

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