As we noted in Section 10.1, the intuitive principle of mathematical induction looks to be a second-order principle that quantifies over numerical properties, and which can't be directly expressed in a first-order theory that only quantifies over numbers. So you might well be wondering: why not work with a second-order arithmetic, rather than hobble ourselves by artificially forcing our formal arithmetic into a first-order straightjacket? True, we now know that – so long as it stays consistent and properly axiomatized – a richer theory won't entirely escape the reach of the Gödel-Rosser Theorem, any more than a first-order theory can. But still, we ought to say at least a little about second-order arithmetics.

Indeed, there is a pressing issue about such theories which really needs to be addressed head on at this point. For if you have done a standard university algebra course, you might very well be feeling pretty puzzled by now. Such a course typically introduces axioms for some version of ‘Second-order Peano Arithmetic’, and there is an elementary textbook proof that these axioms pin down a unique type of structure. But if this second-order arithmetic does pin down the structure of the natural numbers, then – given that any arithmetic sentence makes a determinate claim about this structure – it apparently follows that this theory does enough to settle the truth-value of every arithmetic sentence. Which makes it sound as if there can after all be a (consistent) negation-complete axiomatic theory of arithmetic richer than first-order PA, flatly contradicting the Gödel-Rosser Theorem.