[FOM] Three logical questions around ZF
henri.galinon at libertysurf.fr
Fri Jun 8 17:22:30 EDT 2007
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 ?)
2. Does the tarskian theory of truth for ZF prove any theorem in the
language of ZF that ZF+w-rule doesn't prove ?
[where by "tarskian theory of truth for ZF" we mean : axioms of ZF +
axioms for Satisfaction + sentences of the extended language (ie:
containing "Sat" etc) are allowed to appear in the schemas of ZF.
By "w-rule" we mean an omega-rule (not a finitary one):
F(0), F(1) ...
For all x, [N(x) ---> F(x)]
where N ("natural number"), 1, 2, 3 etc. have been suitably defined
in ZF in one way or another.]
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 ? What about
structures the *complete* second-order theory of which have
isomorphic models only ?
Are there any references on the topic ?
Any reference, information or suggestions are welcome.
PhD Philosophy Student
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