[FOM] Provability in PA

[Studtmann, Paul]

Let I-Sigma-n be Peano Arithmetic (PA) with the induction schema restricted
to Sigma-n formulas.  Can anyone tell me whether the following is true or
false:

(*) There is some n such that for all Sigma-2 sentences, s, if PA |- s, then
I-sigma-n |- s.

I am in fact interested in the general case but (*) is easier to state and
read.

If it is known that (*) is true or that it is false, I would like to know
whether it can be proven to be true, if true, or false, if false, within PA.
And of course I would be interested to know where I can find the proofs.

Paul Studtmann