Well let's say I take as my axioms all true statements about the Natural numbers. Let's take the Goldbach conjecture. Is it an axiom? No one knows because no one knows if it is true or not.
When you say "small agreed upon set" you are in essence talking about a recursively enumerable set of axioms. A collection of axioms that is "small" enough so that one could easily determine if a statement is an axiom.
If you have time for another question, I'm still confused about how it applies to second-order arithmetic, though, since the Peano axioms are well-known and easily listed on a single piece of paper. What difficulty is there be in determining whether a statement is a second-order Peano axiom?
It seems particularly strange since there are apparently fewer axioms than in first-order Peano arithmetic (by replacing an axiom schema with a single induction axiom).
When you say "small agreed upon set" you are in essence talking about a recursively enumerable set of axioms. A collection of axioms that is "small" enough so that one could easily determine if a statement is an axiom.