kurt Gödel's Centenary

The results published by Kurt Gödel during the decade 1929-1939 transformed completely mathematical logic. His doctoral dissertation Uber die Vollstandigkeit des Logikkalculus, of 1929, shows that the provable statements in predicate calculus are exactly those which are true in any interpretation; that is, the logically valid statements. His incompleteness theorem of 1931 competes with that of Pitagoras as one of the better known mathematical results, at least by name, and one of the most frequently mentioned outside of academic mathematics. The demonstration of the consistency of the axiom of choice and the continuum hypothesis, another of his great contributions, was announced in 1938 in the Proceedings of the National Academy of Sciencies, and published in more complete form in 1939.

Gödel published some other papers during those years, but these three contributions to the development of logic are enough to consider him the most outstanding logician of the twentieth century. Gödel did not publish much, but each of his published papers is a jewel of precision and clarity which lets us appreciate the breadth and depth of the author’s thought. It is hard not to fall in the temptation of stating that from the number of his publications and the mechanisms of evaluation of scientific research so much in fashion nowadays, Gödel would hardly have been placed in a very high academic rank, although in the long run the impact of his work would have been recognized.

This is not the appropriate place to analyze in detail Gödel’s mathematical work. However, it is adequate to make a comment on his incompleteness theorem. One of the most surprising aspects of mathematics, and in particular of mathematical logic, is that it is possible to prove the unprovability of some statements. Gödel’s incompleteness theorem established that in any axiomatic system for arithmetic there are statements with the property that neither the statement nor its negation are provable from the axioms of the system. Such statements are said to be undecidable in the system in question. In order to establish the context of the theorem it is convenient to point out that it applies to axiomatic systems that satisfy two more properties; namely, the system must be non-contradictory and algorithmic. An axiomatic system is contradictory if from its axioms a contradiction can be deduced. For mathematics, contradictory systems are not interesting, since from a contradiction any statement can de deduced using the rules of logical deduction: in a contradictory system anything can be proved.

The second of the additional conditions means that there must be an algorithm for determining if a given statement is one of the axioms of the system or not. It is not very useful, for example, a system in which the axioms are all the arithmetical truths, given that, as it is well known, there are many interesting arithmetical statements whose truth status has not been determined. Moreover, the main purpose of an axiomatic system for arithmetic is precisely to serve as a tool to determine the truth of arithmetical statements. Gödel's incompleteness theorem establishes, thus, that in any reasonable axiomatic system for arithmetic, there are arithmetical statements whose truth cannot be determined within the system.

It has been frequently argued that the incompleteness theorem sets some limits to the power of reason. Although it is true that the theorem uncovers certain limitations of the axiomatic method, it is evident that its proof constitutes an admirable intellectual feat which shows the high levels of refinement and sophistication that mathematical reasoning can reach.

In 1940, Gödel was appointed at the Institute of Advanced Studies in Princeton, where he was a colleague and friend of Einstein and von Neumann. He worked there until his death, in January of 1978, due to malnutrition and inanition, product of his neurosis.

This April 26th marks one hundred years since the birth of Kurt Gödel in Brünn, a city of the former Austro-Hungarian province of Moravia. This is an appropriate occasion to point out the importance of his contributions to the development of logic and to the foundations of mathematics. Recently, the fifth and last volume of Gödel’s collected works -published by Oxford- appeared. The series contains all the written material -including previously unpublished work and the correspondence- of this remarkable and singular figure who wrote plenty and published scantly.

Carlos Augusto Di Prisco

Departamento de Matemáticas. Instituto Venezolano de Investigaciones Científicas