We have proven a couple of difficult theorems in this chapter, and by understanding the proof of the Completeness Theorem you have grasped an intricate argument with a wonderful idea at its core. Our results have been directed at structures: What kinds of structures exist? How can we (or can't we) characterize them? How large can they be?

The next chapter begins our discussion of Kurt GĂ¶del's famous incompleteness theorems. Rather than discussing the strength of our deductive system as we have done in the last two chapters, we will now discuss the strength of sets of axioms. In particular, we will look at the question of how complicated a set of axioms must be in order to prove all of the true statements about the standard structure (mathfrak{N}).

In Chapter 4 we will introduce the idea of coding up the statements of (mathcal{L}_{NT}) as terms and will show that a certain set of nonlogical axioms is strong enough to prove some basic facts about the numbers coding up those statements. Then, in Chapters 5 and 6, we will bring those facts together to show that the expressive power we have gained has allowed us to express truth that are unprovable from our set of axioms.

Alternatively, after Chapter 4 you can move straight to Chapter 7 and approach the issue of provability from another direction. But for now, on to Chapter 4!