[FOM] Bryan Ford paper
Randall Holmes
holmes at diamond.boisestate.edu
Thu Apr 29 18:55:55 EDT 2004
Dear FOMers [moderator note that this does contain new content]:
I have read Bryan Ford's paper, which is interesting but does not
prove that ZF is inconsistent (the theory ZF- it discusses is actually
"Zermelo set theory without foundation").
The proof given in this paper is incorrect.
The construction of the "minimal interpretation of ZF- in ZF-" I
believe _is_ correct, in the sense that it does construct a uniquely
determined evaluation of all (codes of) sentences of the language of
ZF- which makes all concrete axioms of ZF- true.
However, it is not possible to prove that all codes for "axioms of
ZF-" are "true" in the minimal interpretation. The possibility which
cannot be ruled out is that there are nonstandard instances of the
axiom of separation which are "false" in the minimal interpretation.
The argument in the second paragraph of p. 21 (which is really not a
full argument, just an illustration) relies on axioms being concretely
given: but there are models of ZF- which contain nonstandard "axioms
of separation", and one cannot carry out the suggested induction on
the (non-well-founded!) structure of a nonstandard "axiom".
Further, consideration of Tarski's theorem shows that the minimal
interpretation must be dishonest in the sense that there must be a
concrete sentence (not just a nonstandard sentence) for which the
minimal interpretation disagrees with the ambient model of ZF- about
its truth value. I think the minimal interpretation will agree with
the ambient model about the truth values of bounded sentences. If the
minimal interpretation did agree with the amibient ZF- about each
concrete sentence, we could consider the predicate informally
expressed by "the nth predicate of natural numbers in L(ZF-) is false
of the natural number n in the minimal interpretation", and apply it
to its own Godel number to get a concrete paradoxical sentence.
I find no support in the paper for the assertion on p. 14 that
existential statements asserted by the minimal interpretation actually
have witnesses (which would conflict with the requirement of
"dishonesty"). It appears clear that nonstandard witnesses to
existential statements can be postulated at any stage of the
construction of the minimal interpretation, with the only requirement
being that one continues to postulate the same nonstandard witnesses
at all later stages.
I believe that the construction of the minimal interpretation given by
Ford is basically similar to the construction one needs to do to prove
the consistency of bounded Zermelo set theory or the theory of types
in Zermelo set theory (which was first done by Kemeny in the 1950's
(?) according to folklore).
--Randall Holmes
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