[FOM] Bryan Ford's paper

[Randall Holmes]

Referring to the paper by Bryan Ford alluded to recently which purports
to prove that ZFC is inconsistent:

This paper is incorrect.  The error I found is on page 21: the author
skims too lightly over the claim that all the (codes for) axioms of
ZF- (the author's ZF- is what everyone else calls "Zermelo set theory
without foundation") are "semantically true" in the sense he defines.
He should have gone through this more carefully: he has not and indeed
cannot show that all (codes for) instances of Separation are
"semantically true" in the sense he defines.

In the second paragraph on p. 21, where he discusses the issue of
"truth" in his defined sense of axioms, he relies on induction on the
concretely given structure of axioms to show that the axioms are
"true" in the internal sense which he defined.  However, since ZF- is
not finitely axiomatizable, it is impossible to rule out the presence
of "codes" for axioms (specifically, instances of separation, which is
the only infinite axiom scheme) on which this induction cannot
actually be carried out, because the "code" doesn't actually code a
concrete axiom (the "axiom" it ostensibly codes is of nonstandard
size).

If it is actually correct that there is a uniquely determined "minimal
interpretation of Zermelo set theory" in Ford's sense which is
definable in Zermelo set theory (I didn't find any mistakes in the
construction, but it is rather complicated) then the final result is
that under the "minimal interpretation" there is a (nonstandard) "axiom
of Zermelo set theory" which is "false" in the minimal interpretation;
in other words, the "minimal interpretation of Zermelo set theory" is
not (in its own terms) an interpretation of Zermelo set theory at all:
it asserts all the concrete axioms of Zermelo set theory, but asserts
the negations of some nonstandard "axioms of Zermelo set theory".

--Randall Holmes