In the preceding chapter we introduced the classes of primitive recursive and recursive functions. In this chapter we introduce the related notions of primitive recursive and recursive sets and relations, which help provide many more examples of primitive recursive and recursive functions. The basic notions are developed in section 7.1. Section 7.2 introduces the related notion of a semirecursive set or relation. The optional section 7.3 presents examples of recursive total functions that are not primitive recursive.

A model of a set of sentences is any interpretation in which all sentences in the set are true. Section 12.1 discusses the sizes of the models a set of sentences may have (where by the size of a model is meant the size of its domain) and the number of models of a given size a set of sentences may have, introducing in the latter connection the important notion of isomorphism. Section 12.2 is devoted to examples illustrating the theory, with most pertaining to the important notion of an equivalence relation. Section 12.3 includes the statement of two major theorems about models, the Löwenheim—Skolem (transfer) theorem and the (Tarski—Maltsev) compactness theorem, and begins to illustrate some of their implications. The proof of the compactness theorem will be postponed until the next chapter. The Löwenheim—Skolem theorem is a corollary of compactness (though it also admits of an independent proof, to be presented in a later chapter, along with some remarks on implications of the theorem that have sometimes been thought ‘paradoxical’).

A normal form theorem of the most basic type tells us that for every formula A there is a formula A* of some special syntactic form such that A and A* are logically equivalent. A normal form theorem for satisfiability tells us that for every set Γ of sentences there is a set Γ* of sentences of some special syntactic form such that Γ and Γ* are equivalent for satisfiability, meaning that one will be satisfiable if and only if the other is. In section 19.1 we establish the prenex normal form theorem, according to which every formula is logically equivalent to one with all quantifiers at the beginning, along with some related results. In section 19.2 we establish the Skolem normal form theorem, according to which every set of sentences is equivalent for satisfiability to a set of sentences with all quantifiers at the beginning and all quantifiers universal. We then use this result to give an alternative proof of the Löwenheim—Skolem theorem, which we follow with some remarks on implications of the theorem that have sometimes been thought ‘paradoxical’. In the optional section 19.3 we go on to sketch alternative proofs of the compactness and Gödel completeness theorems, using the Skolem normal form theorem and an auxiliary result known as Herbrand's theorem. In section 19.4 we establish that every set of sentences is equivalent for satisfiability to a set of sentences not containing identity, constants, or function symbols. Section 19.1 presupposes only Chapters 9 and 10, while the rest of the chapter presupposes also Chapter 12. […]

In the preceding chapter we connected our work on recursion with our work on formulas and proofs in one way, by showing that various functions associated with formulas and proofs are recursive. In this chapter we connect the two topics in the opposite way, by showing how we can ‘talk about’ recursive functions using formulas, and prove things about them in theories formulated in the language of arithmetic. In section 16.1 we show that for any recursive function f, we can find a formula φf such that for any natural numbers a and b, if f(a) = b then ∀y(φf(a, y) ↔ y = b) will be true in the standard interpretation of the language of arithmetic. In section 16.1 we strengthen this result, by introducing a theory Q of minimal arithmetic, and showing that for any recursive function f, we can find a formula ψf such that for any natural numbers a and b, if f (a) = b then ∀y(ψf (a, y) ↔ y = b) will be not merely true, but provable in Q. In section 16.2 we briefly introduce a stronger theory P of Peano arithmetics, which includes axioms of mathematical induction, and explain how these axioms enable us to prove results not obtainable in Q. The brief, optional section 16.3 is an appendix for readers interested in comparing our treatment of these matters here with other treatments in the literature.

By a model of (true) arithmetic is meant any model of the set of all sentences of the language L of arithmetic that are true in the standard interpretation N. By a nonstandard model is meant one that is not isomorphic to N. The proof of the existence of an (enumerable) nonstandard model of arithmetic is as an easy application of the compactness theorem (and the Löwenheim—Skolem theorem). Every enumerable nonstandard model is isomorphic to a nonstandard model ℳ whose domain is the same as that of N, namely, the set of natural numbers; though of course such an ℳ cannot assign the same denotations as N to the nonlogical symbols of L. In section 25.1 we analyze the structure of the order relation in such a nonstandard model. A consequence of this analysis is that, though the order relation cannot be the standard one, it at least can be a recursive relation. By contrast, Tennenbaum's theorem tells us that it cannot happen that the addition and multiplication relations are recursive. This theorem and related results will be taken up in section 25.2. Section 25.3 is a sort of appendix (independent of the other sections, but alluding to results from several earlier chapters) concerning nonstandard models of an expansion of arithmetic called analysis.