[semolinapudding]

There's a bit of a definition issue at play here. When Andreas Blass and Noah Schweber say that there is no proof system for PA_2, they mean that there is no effective proof system that is complete for the full semantics. If you subscribe to their definition of a proof system, you end up saying that there is no such thing as a proof in PA_2, and thus that incompleteness is meaningless -- which I personally find a bit silly.

On the other hand, proof theorists and computer scientists are perfectly happy to use proof systems for second order logic which are not complete. In that case, there are many effective proof systems, and given that the axioms of PA_2 are recursively enumerable (they are in finite number!), Gödel's incompleteness will apply.

If you are still not convinced, I encourage you to decide on a formal definition of what you call PA_2, and what you call a proof in that system. If your proof system is effective, and your axioms are recursively enumerable, then the incompleteness theorem will apply.

[gedpeck]

Thanks for the reply.