Overview

Philosophers, logicians, and mathematicians in the first half of the twentieth century made significant progress toward understanding the connections between mathematics and logic. Major results of these investigations include the Löwenheim-Skolem theorem (1920), Gödel's first and second incompleteness theorems for axiomatic systems (1931), and Henkin's completeness theorem for first-order logics (1949).

Background

Logicians use the terms "formal system," "formal theory," "formal language," and "logic" almost interchangeably to refer to any set K whose members are in practice subject to rules of inference. Each formal system has three components: grammar, deductive system, and semantics, which together determine the logic of K. The grammar consists of the members of K, such as symbols, logical operators, constants, variables, etc., and the rules governing the formation of terms. A "wellformed formula" (known as a wff) is any statement made according...