Elliott Mendelson

The Propositional Calculus

Sentences may be combined in various ways to form more complicated sentences. Let us consider only truth-functional combinations, in which the truth or falsity of the new sentence is determined by the truth or falsity of its component sentences.

Formal Number Theory

Together with geometry, the theory of numbers is the most immediately intuitive of all branches of mathematics. It is not surprising then that attempts to formalize mathematics and to establish a rigorous foundation for mathematics should begin with number theory. The first semiaxiomatic presentation of this subject was given by Dedekind in 1879 and has come to be known as Peano’s Postulates.* It can be formulated as follows: n n(P1) n nO is a natural number. n n n n n(P2) n nIf x is a natural number, there is another natural number denoted by x′ (and called the successor of x).† n n n n n(P3) n nO ≠ x′ for any natural number x. n n n n n(P4) n nIf x′ = y′ then x = y. n n n n n(P5) n nIf Q is a property that may or may not hold for natural numbers, and if (I) 0 has the property Q and (II) whenever a natural number x has the property Q, then x′ has the property Q, then all natural numbers have the property Q (Principle of Induction).

Axiomatic Set Theory

A prime reason for the increase in importance of mathematical logic in this century was the discovery of the paradoxes of set theory and the need for a revision of intuitive (and contradictory) set theory. Many different axiomatic theories have been proposed to serve as a foundation for set theory, but, no matter how they may differ at the fringes, they all have as a common core the fundamental theorems that mathematicians need in their daily work. A choice among the available theories is primarily a matter of taste, and we make no claim about the system we shall use except that it is an adequate basis for present-day mathematics.

A note on the Axioms of Restriction and Fundierung

Abstract A sentence (W) is found such that (Res), the Axiom of Restriction, is equivalent to the conjunction of (F), the axiom of Fundierung, and (W), and furthermore (W) is independent of (F). The discussion takes place in NBG, the von Neumann-Bernays-Godel set theory.