Many examples are now known of sentences undecidable in Peano's Arithmetic (PA) assuming that PA is consistent. Are some or all of these sentences also undecidable in Second Order Arithmetic (SOA) if SOA is consistent? I am interested in this question because SOA is an axiomatizable theory even though it contains some of the axioms of second order logic-whose set of universally valid sentences is not recursively enumerable. If s is a sentence undecidable in SOA which asserts that all positive integers have a certain property, what could we say about the smallest positive integer n that is asserted by the negation of s to fail to have this property? There are models of PA which contain n as an "infinite" positive integer. But are there any models of SOA which contain "infinite" positive integers? It seems that I do not fully understand the model theory of SOA. 



A question about sentences undecidable in Peano's Arithmetic
Since you're asking about provability in SOA, you're presumably referring to a standard deduction system, such as the one in Steve Simpson's book, which is sound not only for the standard (or "full") semantics but also for Henkin semantics. 


The most natural example of a sentence undecidable in SOA is "SOA is consistent". It's unprovable because of Gödel's second incompleteness theorem, and it's not refutable because it's true.


Gödel's second incompleteness theorem and his completeness theorem together imply that some nonstandard models of SOA contain nonstandard integers n satisfying the formula that expresses "n is the Gödel number of a proof of a contradiction on SOA.


