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The arithmetization of syntax

In the main part of this chapter, we introduce Godel's simple but wonderfully powerful idea of associating the expressions of a formal theory with code numbers. In particular, we will fix on a scheme for assigning code numbers first to expressions of L A and then to proof-like sequences of such expressions. This coding scheme will correlate various syntactic properties with purely numerical properties – in a phrase, the scheme arithmetizes syntax . For example, take the syntactic property of being a term of L A . We can define a corresponding numerical property Term , where Term ( n ) holds just when n codes for a term. Likewise, we can define Atom ( n ), Wff ( n ), and Sent ( n ) which hold just when n codes for an atomic wff, a wff, or a closed wff (sentence) respectively. It will be easy to see – at least informally – that these numerical properties are primitive recursive ones. More excitingly, we can define the numerical relation Prf ( m, n ) which holds just when m is the code number in our scheme of a PA-derivation of the sentence with number n . It will also be easy to see – still in an informal way – that this relation too is primitive recursive. The short second part of the chapter introduces the idea of the diagonalization of a wff. This is basically the idea of taking a wff φ (y), and substituting (the numeral for) its own code number in place of the free variable. We can think of a code number as a way of referring to a wff.