8 - Two formalized arithmetics

Summary

We now move on from the generalities of the previous chapters, and look at some particular formal arithmetics. In this chapter, we limber up by looking at Baby Arithmetic, and then we start exploring Robinson Arithmetic. Later, in Chapter 10, we'll be introducing Peano Arithmetic, the strongest of our initial range of formal arithmetics.

These theories differ in strength, but they do share one key feature: the theories' deductive apparatus is no richer than familiar first-order logic. So we can quantify, perhaps, over all numbers: but our theories will lack second-order quantifiers, i.e. we can't quantify over all numerical properties.

BA, Baby Arithmetic

We begin with a very simple theory which ‘knows’ about the addition of particular numbers, ‘knows’ its multiplication tables, but can't express general facts about numbers at all (it lacks the whole apparatus of quantification). Hence our label Baby Arithmetic, or BA for short. As with any formal theory, we need to characterize (a) its language, (b) its deductive apparatus, and (c) its axioms.

(a) BA's language is L_B = 〈ℒ_B, I_B〉. ℒ_B's non-logical vocabulary is the same as that of ℒ_A (Section 4.3): hence there is a single individual constant ‘0’, the one-place function symbol 's’, and the two-place function symbols ‘+’ and ‘×’. So ℒ_B contains the standard numerals. However, ℒ_B's logical apparatus is restricted. As we said, it lacks quantifiers and variables.