Abstract
Publisher Summary This chapter discusses the classical and the ω-complete arithmetic. It deals with two formal systems for the theory of natural numbers, both of which are applied second-order functional calculi with equality and the description operator. The two systems have the same primitive symbols, rules of formation, and axioms, differing only in the rules of inference. The classes of number formulas (nfs) and propositional formulas (pfs) are defined inductively as the least classes of formal expressions. Together the nfs and pfs are called formulas. pf containing no free variables is called closed, and a formula in which no quantifier binding a function variable occurs is called elementary. The chapter provides a description of semantics after making meta-mathematical observation that the notions and axioms which are assumed in meta-mathematics comprise all the ordinary notions and axioms of set theory, so that such non-elementary notions can be used in definitions as classes and functions of arbitrarily high types.
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