Symbolic logic, as developed by such nineteenth-century pioneers as George Boole, is one of the cardinal achievements of modern mathematics. Indeed, contemporary computer technology would scarcely be conceivable without the kind of foundation provided by Boolean algebra. Yet the tool of symbolic logic has critical limitations and can easily be misapplied, particularly if it is interpreted as a system for manipulating strings of symbols without regard to their semantic referents—that is, strings divorced from real-world meaning.

The shortcomings of a string-manipulation approach to mathematics and logic became fully evident only around 1900, when it led to a number of paradoxes (apparent contradictions), which appeared simultaneously in several branches of the field, seeming to threaten the very foundations of mathematics. The crowning blow came in 1931, when Kurt Gödel proved rigorously that for any system based on such formal manipulations of symbols from a finite set of postulates, either

1. the system would be too simple to decide the truth or falsehood of all statements of arithmetic; or else

2. the system's postulates would eventually generate a contradiction.

While some of these paradoxes are quite sophisticated, their flavor is conveyed by the following simple and familiar example, which can be followed by anyone who has a rudimentary acquaintance with mathematical symbolism.