[FOM] arithmetical soundness of ZFC
nweaver at math.wustl.edu
Sun May 17 11:10:04 EDT 2009
Fred Muller and I have been corresponding off-list about
a paper of his, "The implicit definition of the set-concept"
(Synthese 138 (2004), 417-451). With his permission I'd
like to move the discussion to the FOM list because it
involves an issue of general interest that I think is not
so well appreciated.
Fred does an excellent job, much better than I could, of
expressing some of the philosophical difficulties with
classical set theory. His response, as I understand it,
is to treat the set concept as being implicitly defined by
(say) the formal system ZFC. So we can let ourselves off
the hook on the problem of having to identify exactly what
sets are before we adopt the formal theory.
The paper posits a pair of requirements that need to be
met in order for an implicit definition of a concept to
be successful: "(Cr1) it is logically possible that the
axioms are true and (Cr2) they save the paradigm linguistic
phenomena". (Cr1) is defended in the case of set theory
by the argument that "ZF has been explored almost to the
point of exhaustion ... so that if it were to harbor some
contradiction, it would have been found by now."
The problem I have with this is that it only argues for ZFC
being "possibly true" in the weak sense of being consistent.
But mere consistency does not automatically entail that ZFC
is arithmetically valid. Is every sentence of first order
number theory that is provable in ZFC actually true? For
example, if ZFC proves that Turing machine x halts on input
y for some specific values of x and y, is this actually the
I have the impression that it is fairly common for
philosophical defenses of set theory to miss this point, and
to assume that the only issue is the question of consistency.
I am not just playing devil's advocate. I think it is
quite likely that ZFC is consistent but not arithmetically
sound. I spell out this objection in more detail in
Section 2 of my "indispensable" paper (announced in FOM
post # 013607).
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