[FOM] Three logical questions around ZF

Aatu Koskensilta aatu.koskensilta at xortec.fi
Tue Jun 12 11:48:52 EDT 2007

Henri Galinon writes:

  Dear FOMers,

  1. Can we decide any interesting set-theoretic hypothesis in
  (iterated) tarskian truth-theoretic extensions of ZF ? (If not,
  could someone give the flavour of why it can't be so ?)

It seems not. Let kappa be an inaccessible, and interpret the set
quantifiers as ranging over V_kappa, and truth as truth-in-V_kappa,
and you get a model of any extension of ZFC with an iterated truth
predicate. No particularly interesting set theoretic hypotheses or
theorems are known to follow merely from the existence of an
inaccessible (and you don't get even that by iterating truth).

  2. Does the tarskian theory of truth for ZF prove any theorem in the
  language of ZF that ZF+w-rule doesn't prove ?

Yes. For example: "for all P, if ZF proves 'P holds in V_omega+omega'
then P holds in V_omega+omega'.

  3. Something a bit different. Using second-order logic, we can give
  a categoric finite axiomatization of arithmetic (Second-oder PA does
  the job). What other "important" structures (eg models of ZF ?) are
  categorically axiomatizable in second-order logic ?

Second order set theory is "almost categorical"; given any two models
of second order set theory, one is an initial segment of the
other. Practically all important mathematical structures in ordinary
mathematics are second order axiomatizable, and in fact Pi-1-1
axiomatizable. In set theory we know of exceptions, of course, and
e.g. being a measurable is not a second order property.

A nice source of information about second order logic is Stewart
Shapiro's _Foundations without Foundationalism -- a Case for
Second-Order Logic_.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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