Kurt Gödel

Austrian American mathematician

• Born: April 28, 1906; Brünn, Austria-Hungary (now Brno, Czech Republic)
• Died: January 14, 1978; Princeton, New Jersey

Kurt Gödel did fundamental work in many areas of mathematical logic and made several contributions to philosophy and physics. His most famous achievement was the enunciation and proof of the incompleteness theorems of arithmetic, the consequences of which cut across all branches of mathematics and gave rise to results in computer science as well. Mathematical logic assumed a more central position in mathematics following Gödel’s career.

Primary field: Mathematics

Specialty: Mathematical analysis

Early Life

Kurt Gödel was born in Brünn, Moravia, a territory of the Austro-Hungarian Empire, on April 28, 1906, to Rudolf Gödel and Marianne Handschuh. The Gödels were part of the German-speaking minority in the region, and although Moravia would become a part of Czechoslovakia after World War I, Gödel’s family retained its German-based linguistic and cultural identity. As a factory owner, Rudolf’s financial circumstances were such that he could afford a good education for his son, who was a distinguished scholar even as a child. However, Gödel suffered from rheumatic fever and bore the effects of the disease for the rest of his life. Later, he also suffered from hypochondria.

In 1916, Gödel enrolled at a nearby gymnasium, or secondary school, where he developed an interest in mathematics and physics. He then entered the University of Vienna, an academic center for the German-speaking community of the newly established Czechoslovakia. In Vienna, Gödel renounced his Czech citizenship and became an Austrian citizen. He studied physics, philosophy, and mathematics and took part in the meetings of the Vienna Circle, a group of philosophers and scientists who debated major scientific issues of the day. Gödel’s involvement in the Vienna Circle put him at the forefront of current developments in the application of logic to science and other areas.

Life’s Work

Gödel’s doctoral dissertation, submitted in 1929, demonstrated the completeness of first-order logic. In mathematics, first-order logic examines statements referring to collections of objects, called sets, but not collections of sets. Completeness refers to the match between the true statements within an area and the statements that are provable from the axioms, or starting points. Demonstrating completeness involves coming up with a technique for showing how any true statement can be proved.

In the early 1930s, a popular view of mathematics held that the field was really about what was provable from a given set of axioms, equating truth with provability. Much to the surprise of the advocates of that view, Gödel was able to demonstrate the incompleteness of a subject as fundamental as arithmetic. He accomplished this by translating general statements into arithmetic, then combining facts about arithmetic with a variant on what is known as the liar paradox, which points out the difficulty of determining whether the statement “This statement is false” is true or false. Gödel translated the statement “This statement is not provable” into his arithmetic, from which he concluded that there was indeed a true statement that was not provable. He was able to extend his result to general conclusions about the inability of a system of axioms to demonstrate its own freedom from contradiction.

Although it took time for the mathematics community at large to appreciate the importance of Gödel’s work, mathematical logicians around the world came to recognize the significance of his theorems. In particular, Oswald Veblen, an American logician who was involved in the creation of the Institute for Advanced Study in Princeton, New Jersey, invited Gödel to the United States to discuss his work. At Princeton, Gödel was able to conduct research without any teaching obligations. This served him well, as he had never been a particularly dedicated or effective instructor. For much of the 1930s, Gödel traveled between Vienna and Princeton, where he worked with logicians and computer scientists like John von Neumann and Alan Mathison Turing. However, various bouts of ill health and psychological trouble occasionally interfered.

Despite increasing political tension in Austria, Gödel continued to return to Vienna. Although he had no Jewish ancestry, the Vienna Circle had been associated with the political Left, and Gödel was deemed untrustworthy by the government. In 1938, he married Adele Porkert Nimbursky, a dancer whom he had known for a number of years. His family was not enthusiastic about the marriage. Two years later, when he found himself in Vienna after the outbreak of World War II, Gödel and his wife relocated to the United States via the Soviet Union. Gödel never returned to Europe.

One of the statements of greatest interest to logicians of the era had been formulated by German mathematician Georg Cantor, who founded set theory in the nineteenth century. Cantor’s continuum hypothesis states that there are no infinite sets whose size is between that of the whole numbers and that of the real numbers. Gödel succeeded in demonstrating that this hypothesis, and a generalized version of it, could be added to the standard axioms of set theory without contradiction. Although many mathematicians were not convinced that the continuum hypothesis was true, the techniques that Gödel introduced to demonstrate his result were used to look at other questions about sets.

After Gödel relocated permanently to Princeton’s Institute for Advanced Study, the focus of his work shifted to questions about philosophy and physics. His proof of the incompleteness theorem had not settled issues about the foundations of mathematics, and Gödel’s point of view remained controversial. He believed that one could have a special kind of “perception” of mathematical objects, much as one can see physical objects.

In addition to his philosophical pursuits, Gödel also spent time discussing cosmology with physicist Albert Einstein. Much of Gödel’s work from his later career was not published during his lifetime. He grew increasingly reclusive and suspicious of the outside world in general. Although he received honorary degrees from Harvard and Yale, he did not accept other honors and awards for fear of traveling or being in public.

After retiring from the institute in 1976, Gödel and his wife were faced with mounting medical problems, and Gödel became increasingly unstable. When his wife was hospitalized, he refused to eat food prepared by anyone else. He died in 1978.

Impact

No other mathematician had as much influence on the mathematics and philosophy of the twentieth century as did Gödel. The techniques he introduced to establish his incompleteness theorem became standard for logicians, while his work in set theory laid the foundation for subsequent achievements in the field. Gödel’s theories in physics and the philosophy of mathematics were debated in both technical and nontechnical settings, and the popularization of his ideas extended to connections with art, music, and literature.

Mathematician Paul J. Cohen investigated the problem of the continuum hypothesis and demonstrated in 1963 that the negation of that hypothesis could also be added to the standard axioms of set theory without contradiction. When this was combined with Gödel’s earlier thinking, the question of how to decide the “truth” of axioms of set theory was rendered more puzzling. Many questions about the philosophy of mathematics had been raised before Gödel’s time without expectation of a definite answer, but Gödel demonstrated that mathematics has the technical resources to answer questions about its fundamental principles. Because of his work, mathematical logic moved closer to the center of the stage of intellectual inquiry.

Bibliography

Davis, Martin. Engines of Logic: Mathematicians and the Origin of the Computer. New York: Norton, 2000. Print. A view of Gödel’s work from the standpoint of the development of computer science.

Franzén, Torkel. Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley: Peters, 2005. Print. Attempts to clarify the technical content of the incompleteness theorem and its misapplications in philosophy.

Gödel, Kurt. Collected Works. New York: Oxford UP, 1986–2003. Print. Includes five volumes of papers with introductions helpful for all readers.

Nagel, Ernest, and James R. Newman. Gödel’s Proof. New York: New York UP, 2002. Print. Revised edition of the classic exposition of the ideas in Gödel’s paper for those with limited mathematical experience.