[FOM] Question

[Rupert McCallum]

Woodin defines a large cardinal axiom as follows.

"(exists x) phi" is a _large_cardinal_axiom_ if 
       1. phi(x) is a Sigma_2 formula 
       2. (As a theorem of ZFC) if kappa is a cardinal such that 
              V |= phi[kappa] 
          then kappa is strongly inaccessible, and for all 
          partial orders P \in V_kappa 
              V^P |= phi[kappa]."

Let T be the set of all sentences in the language of second-order
arithmetic which are either provable in first-order logic or which are
provable in an extension of ZFC by finitely many large-cardinal axioms
which are all true in V_kappa for some inaccessible kappa. T is a
consistent theory, and assuming two inaccessibles, it contains every
true sentence in the first-order language of arithmetic.

Does T contain every true sentence in the second-order language of
arithmetic? 



 
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