The system Aoo

In this chapter we construct the formal system Aoo. It is a very simple arithmetic with familiar fundamental concepts. These are: the natural number zero, the successor function, the operation of repeatedly applying a function, the operation of forming functions by abstraction, equality and inequality between numerical expressions, and the logical connectives, conjunction and disjunction. The atomic statements are equations and inequations between numerical terms, compound statements are built up from atomic statements by conjunction and disjunction. Negation, material implication and material equivalence are definable, but existential quantification and universal quantification are unrepresentable.

We give definitions of Aoo-truth and of Aoo-falsity for closed Aoo-statements, and show that they are exclusive properties. We also show that a closed Aoo-statement is Aoo-true if and only if it is an Aoo-theorem. Thus the system Aoo is consistent in the sense that Aoo-theorems are Aoo-true; and is complete in the sense that Aoo-true Aoo-statements are Aoo-theorems. We give a procedure which applied to a closed Aoo-statement will terminate and tell us whether it is Aoo-true or is Aoo-false. Thus the system Aoo is decidable.

The Aoo-rules of formation

To construct the system Aoo we first list the Aoo-signs and attach a type to each proper Aoo-symbol and give each Aoo-symbol a name which will assist the reader in understanding how the system was first conceived. Parentheses round type symbols are usually omitted by association to the left as explained in Ch. 1. (See table overleaf.)