Entscheidungsproblem: What’s in a Word?

In 1900, the celebrated German mathematician David Hilbert (1862–1943), professor of mathematics in the University of Göttingen, delivered a lecture at the International Mathematics Congress in Paris in which he listed 23 significant “open” (mathematicians’ jargon for “unsolved”) problems in mathematics. Hilbert’s second problem was: Can it be proved that the axioms of arithmetic are consistent? That is, that theorems in arithmetic, derived from these axioms, can never lead to contradictory results? To appreciate what Hilbert was asking, we must understand that in the fin de siècle world of mathematics, the “axiomatic approach” held sway over mathematical thinking. This is the idea that any branch of mathematics must begin with a small set of assumptions, propositions, or axioms that are accepted as true without proof. Armed with these axioms and using certain rules of deduction, all the propositions concerning that branch of mathematics can be derived as theorems. The sequence of logically derived steps leading from axioms to theorems is, of course, a proof of that theorem. The axioms form the foundation of that mathematical system. The axiomatic development of plane geometry, going back to Euclid of Alexandria (fl . 300 BCE ) is the oldest and most impressive instance of the axiomatic method, and it became a model of not only how mathematics should be done, but also of science itself. Hilbert himself, in 1898 to 1899, wrote a small volume titled Grundlagen der Geometrie (Foundations of Geometry) that would exert a major influence on 20th-century mathematics. Euclid’s great work on plane geometry, Elements, was axiomatic no doubt, but was not axiomatic enough. There were hidden assumptions, logical problems, meaningless definitions, and so on. Hilbert’s treatment of geometry began with three undefined objects—point, line, and plane—and six undefined relations, such as being parallel and being between. In place of Euclid’s five axioms, Hilbert postulated a set of 21 axioms. In fact, by Hilbert’s time, mathematicians were applying the axiomatic approach to entire branches of mathematics.