Peano arithmetic (PA, or arithmetic, for short) is classical first-order arithmetic with induction. The aim of this chapter is to define the concepts mentioned in, and describe the proofs of, five important theorems about Bew(x), the standard “provability” or “theoremhood” predicate of PA:

1. (i) If ⊢ S, then ⊢ Bew(⌜S⌝),

2. (ii) ⊢ Bew(⌜(S → T)⌝) → (Bew(⌜S⌝) → Bew(⌜T⌝)),

3. (iii) ⊢ Bew(⌜S⌝) → Bew(⌜Bew(⌜S⌝)⌝),

4. (iv) Bew(⌜S⌝) is a Σ sentence, and

5. (v) if S is a Σ sentence, then ⊢ S → Bew(⌜S⌝)

(for all sentences S, T of Peano arithmetic).

‘⊢’ is, as usual, the sign for theoremhood; in this chapter we write ‘⊢S’ to mean that S is a theorem of PA. ┌S┐ is the numeral in PA for the Gödel number of sentence S, that is, if n is the Gödel number of S, then ┌S┐ is 0 preceded by n occurrences of the successor sign s. Bew(┌S┐) is therefore the result of substituting ┌S┐ for the variable x in Bew(x), and (iii) immediately follows from (iv) and (v). Bew(┌S┐) may be regarded as a sentence asserting that S is a theorem of PA. Σ sentences (often called Σ1 sentences) are, roughly speaking, sentences constructed from atomic formulas and negations of atomic formulas by means of conjunction, disjunction, existential quantification, and bounded universal quantification (“for all x less than y”), but not negation or universal quantification. A precise definition is given below.

Notice the distinction between ‘Bew(x)’ and ‘⊢’. ‘Bew(x)’ denotes a certain formula of the language of PA and thus Bew(x) is that formula; it is a formula that is true of (the Gödel numbers of) those formulas of PA that are provable in PA.