Back in Chapter 10, we introduced the weak arithmetic Q, and soon saw that it is boringly incomplete. In Chapter 12, the stronger arithmetic Δ0 was defined, and this too can be seen to be incomplete without invoking Gödelian methods. Then in Chapter 13 we introduced the much stronger first-order theory PA, and remarked that we couldn't in the same easy way show that it fails to decide some elementary arithmetical claims. However, in the last chapter it has turned out that PA also remains incomplete.

Still, that result in itself isn't yet hugely exciting, even if it is perhaps rather unexpected (see Section 13.3). After all, just saying that a particular theory T is incomplete leaves wide open the possibility that we can patch things up by adding an axiom or two more, to get a complete theory T+. As we said at the very outset, the real force of Gödel's arguments is that they illustrate general methods which can be applied to any theory satisfying modest conditions in order to show that it is incomplete. They reveal that a theory like PA is not only incomplete but in a good sense incompletable.

The present chapter explains these crucial points.

Generalizing the semantic argument

In Section 21.3, we showed that PA is incomplete on the semantic assumption that its axioms are true (given that its standard first-order logic is truth-preserving). In this section, we are going to extend the semantic argument for incompleteness to other theories.