As we noted at the end of Chapter 9, it is rather natural to suggest that the intuitive principle of arithmetical induction should be regimented as a secondorder principle that quantifies over numerical properties, and which therefore can't be directly expressed in a first-order theory that only quantifies over numbers. So why not work with a second-order theory, rather than hobble our formal arithmetic by forcing it into a first-order straightjacket?

True, we have discovered that – so long as it stays consistent and effectively axiomatized – any theory containing enough arithmetic will be incomplete. But still, we ought to say at least a little about second-order arithmetics, and this is as good a place as any. Indeed, if you have done a university mathematics course, you might very well be feeling rather puzzled by now. Typically, at some point, you are introduced to axioms for a version of ‘Second-order Peano Arithmetic’ and are given the elementary textbook proof that these axioms are categorical, i.e. 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.