The previous chapter concerned axiomatized formal theories in general. This chapter introduces some key concepts we need in describing formal arithmetics in particular, notably the concepts of expressing and capturing numerical properties. But we need to start with two quick preliminary sections, about notation and about the very idea of a property.

Three remarks on notation

(a) Gödel's First Incompleteness Theorem is about the limitations of axiomatized formal theories of arithmetic: if a theory T is consistent and satisfies some other fairly minimal constraints, we can find arithmetical truths that can't be derived in T. Evidently, in discussing Gödel's result, it will be very important to be clear about when we are working ‘inside’ some specified formal theory T and when we are talking informally ‘outside’ that particular theory (e.g. in order to establish truths that T can't prove).

However, we do want our informal talk to be compact and perspicuous. Hence we will tend to borrow the standard logical notation from our formal languages for use in augmenting mathematical English (so, for example, we might write ‘∀x∀y(x + y = y + x)’ as a compact way of expressing the ‘ordinary’ arithmetic truth that the order in which you sum numbers doesn't matter).