[FOM] Three logical questions around ZF

[henri galinon]

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 ?)

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.

Best,

Henri


PhD Philosophy Student
IHPST, Paris