Kurt Gödel

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Kurt Gödel
Born

April 28 1906(1906-04-28)
Brünn (Brno) Austria-Hungary
Died January 14 1978 (aged 71)

Princeton, New Jersey, U.S.
Field Mathematics
Institutions Institute for Advanced Study
Alma mater University of Vienna
Academic advisor  Hans Hahn
Known for Gödel's incompleteness theorems
Notable prizes Albert Einstein Award (1951)

Kurt Gödel (April 28, 1906 – January 14, 1978) was one of the most significant logicians of all time, whose work had an immense impact on 20th century philosophy, logic, and mathematics.

He is best known for his two incompleteness theorems, which he published in 1931 at 25 years of age, a year after finishing his doctorate at the University of Vienna. The theorems and the groundbreaking techniques they employed forced a rethinking of the foundations of mathematics, and initiated the field of metamathematics. Gödel went on in his mathematical work to establish important theorems in set theory and to clarify the connections between classical logic, intuitionistic logic, and modal logic.

Gödel also made direct contributions to philosophy, and was influenced in his thinking by the writings of Immanuel Kant, Edmund Husserl, and, perhaps most strongly, Gottfried Leibniz. He defended mathematical realism and, though he did not make his religious views public, was critical of materialism and described himself as a philosophical theist. Gödel recorded an unpublished revision of the Ontological Argument for the existence of God that makes use of modern advances in modal logic, and which has spawned much discussion among logicians and philosophers since its publication after his death.

Life

Childhood

Kurt Friedrich Gödel was born April 28, 1906, in Brno (German: Brünn), Moravia, Austria-Hungary (now the Czech Republic) into the ethnic German family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh). At the time of his birth the town had a slight German-speaking majority and this was the language of his parents.

He automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian Empire broke up at the end of World War I. He later told his biographer John W. Dawson that he felt like an "exiled Austrian in Czechoslovakia" ("ein österreichischer Verbannter in Tschechoslowakien") during this time. He spoke very little Czech. He became an Austrian citizen by choice at age 23. When Nazi Germany annexed Austria, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became an American citizen.

In his family, young Kurt was known as Der Herr Warum ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage.

He attended German language primary and secondary school in Brno and completed them with honors in 1923. Although Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna to go to medical school at the University of Vienna (UV). During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, and the writings of Immanuel Kant.

Studying in Vienna

At the age of 18, Kurt joined his brother Rudolf in Vienna and entered the UV. By that time, he had already mastered university-level mathematics. Although initially intending to study theoretical physics, Kurt also attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism. He read Kant's Metaphysische Anfangsgründe der Naturwissenschaft, and participated in the Vienna Circle with Moritz Schlick, Hans Hahn, and Rudolf Carnap. Kurt then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, Kurt became interested in mathematical logic.

Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik (Principles of Theoretical Logic), an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system? This was the topic chosen by Gödel for his doctorate work. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, Gödel established the completeness of the first-order predicate calculus (this result is known as Gödel's completeness theorem). He was awarded the doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science.

Working in Vienna

In 1931, Gödel published his famous incompleteness theorems in "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" (called in English "On formally undecidable propositions of Principia Mathematica and related systems"). In that article, he proved that for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or ZFC), then:

  1. If the system is consistent, it cannot be complete. (This is generally known as the incompleteness theorem.)
  2. The consistency of the axioms cannot be proved within the system.

These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.

In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the fact that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement. That is, for any humanly constructible set of axioms for arithmetic, there is a formula which obtains in arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to solve several technical issues, such as encoding statements, proofs, and the very concept of provability into the natural numbers. He did this using a process known as Gödel numbering.

Gödel earned his habilitation at the UV in 1932, and in 1933 he became a Privatdozent (unpaid lecturer) there. Hitler's 1933 ascension in Germany had little effect on Gödel in Vienna, as he took little interest in politics. He was, however, much affected by the 1936 murder of Moritz Schlick (whose seminar had aroused Gödel's interest in logic) by a deranged student, which resulted in Gödel's first nervous breakdown.

Visits to the USA

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend. He delivered an address to the annual meeting of the American Mathematical Society. During this year, Gödel also developed the ideas of computability and recursive functions to the point where he delivered a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using Gödel numbering.

In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his Ph.D. at Princeton, took notes of these lectures which have been subsequently published.

Gödel would visit the IAS again in the autumn of 1935. The traveling and the hard work had exhausted him and the next year he had to recover from a depression. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the axiom of choice and of the continuum hypothesis; he would go on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.

He married Adele Nimbursky (née Porkert), whom he had known for over ten years, on September 20, 1938. Their relationship had been opposed by his parents on the grounds that she was a divorced dancer, six years older than he. They had no children.

Subsequently, he left for another visit to the USA, spending the autumn of 1938 at the IAS and the spring of 1939 at the University of Notre Dame.

Princeton

After the Anschluss in 1938, Austria had become a part of Nazi Germany. Germany abolished the title of Privatdozent, so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him. His predicament precipitated when he was found fit for military service and was now at risk of being conscripted into the German army. World War II started in September 1939. In January 1940, Gödel and his wife left Europe. Due to the difficulty of an Atlantic crossing, they took the trans-Siberian railway and passed through Japan en route to the U.S.. Arriving in San Francisco, California on March 4, 1940, they crossed the U.S. by train so that Gödel could take up a position at the IAS in Princeton, New Jersey.

Gödel very quickly resumed his mathematical work. In 1940, he published his work Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory which is a classic of modern mathematics. In that work he introduced the constructible universe, a model of set theory in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are true in the constructible universe, and therefore must be consistent with the Zermelo-Frankel axioms for set theory (ZF). Paul Cohen later constructed a model of ZF in which AC and GCH are false; together these proofs mean that AC and GCH are independent of the ZF axioms for set theory.

During his many years at the Institute, Gödel's interests turned to philosophy and physics. He studied and admired the works of Gottfried Leibniz, but came around to the (unsupported) belief that most of Leibniz's works had been suppressed. To a lesser extent he studied Kant and Edmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's ontological proof of God's existence. This is now known as Gödel's ontological proof.

In the late 1940s, Gödel demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in general relativity. These "rotating universes" would allow time travel and caused Einstein to have doubts about his own theory.

Gödel became a permanent member of the IAS in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976.

Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award, in 1951, and was also awarded the National Medal of Science, in 1974.

Death

In later life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he wouldn't eat unless his wife, Adele, tasted his food for him. Late in 1977, Adele was hospitalized for six months and could not taste Gödel's food anymore. In her absence, he refused to eat, eventually starving himself to death. He was 65 pounds when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978.[1]

Works

Completeness Theorem

Gödel proved the Completeness Theorem for his doctoral dissertation, submitted to the University of Vienna in 1929. It was undertaken in response to a question first posed by David Hilbert: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system? Gödel's result yields a positive answer, stating that in first-order logic every logically valid formula is provable. The Completeness Theorem establishes a fundamental connection between proof theory and model theory, giving a link between semantics and syntax. It should not, however, be misinterpreted as collapsing the distinction between proof and truth; in fact, Gödel later showed that there are inherent limitations to what can be achieved with formal proofs in mathematics. The Completeness Theorem is a central property of first-order logic, but it does not hold for all logics or theories. In particular, it does not apply to second-order logic, or to theories powerful enough to prove basic truths of arithmetic, as Gödel went on to prove in his incompleteness theorems.

Incompleteness Theorems

Gödel's incompleteness theorems, proved by Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. The theorems are of considerable importance to the philosophy of mathematics, where they are widely regarded as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible. More controversially, some philosophers argue that the theorems imply the mind cannot be modeled by a computer, and thus provide a refutation of mechanism.

The First Incompleteness Theorem appeared as Proposition VI in Gödel's 1931 paper On Formally Undecidable Propositions in Principia Mathematica and Related Systems. Perhaps the single most celebrated result in mathematical logic, it states (in altered form) that:

For any consistent formal theory T that proves basic arithmetical truths, an arithmetical statement that is true but not provable in the theory can be constructed. That is, any theory capable of expressing elementary arithmetic cannot be both consistent and complete.

(Here, "theory" refers to an infinite set of statements, some of which are axioms, and others that are taken as true because they are implied by the axioms. "Provable in the theory" means "derivable from the axioms and primitive notions of the theory, using standard first-order logic." A theory is "consistent" if it never proves a contradiction. "Can be constructed" means that some mechanical procedure exists which can construct the statement, given the axioms, primitives, and first order logic. "Elementary arithmetic" consists merely of addition and multiplication over the natural numbers.)

Roughly speaking, to prove the First Incompleteness Theorem, Gödel described a method to construct for any formal system a statement, G, that asserts: "G cannot be proved in T." If G were able to be proved under T's axioms, then T would have a theorem, G, which contradicts itself, and thus T would be inconsistent. But if G were not provable, then it would be true (for G expresses this very fact) and thus T would be incomplete. Statements such as G, known as Gödel sentences, provide counterexamples to completeness, and are constructible for any given formal system powerful enough to capture arithmetic statements.

To construct such self-referential sentences, Gödel invented a technique to represent statements as numbers. Using this technique, it is possible to encode any given statement R as some number Z (e.g. "23987548"). Since the given formal system is powerful enough to prove statements about the natural numbers, the system can "talk about" its own statements by placing the number Z (which represents a statement) as an argument in a formula. Statements such as the Gödel sentence, which talk about the provability or unprovability of other statements, can therefore be represented as formulas taking coded numbers as arguments. To construct a Gödel sentence, we take the statement that says, "The formula with Gödel number x is unprovable in T." Using the Gödel numbering technique, this statement can itself be coded into a Gödel number, z. By substituting z for x, we arrive at a formula that essentially says "I am unprovable in T." As outlined in the informal proof above, this sentence provides a counterexample to T's completeness.

Gödel's Second Incompleteness Theorem, which was published in the same 1931 paper, can be stated as follows:

Within any formal consistent theory T that proves basic arithmetical truths, the consistency of T cannot be proved.

The theorem builds on the result obtained in the earlier theorem. According to the First Incompleteness Theorem, if T is consistent, then there is a sentence that is true but unprovable in T. Namely, this is the sentence G, which says "G is unprovable in T." We might state this result as the following conditional: if T is consistent, then G is unprovable in T. Gödel realized that it follows that if we have a proof of the consistency of T, by implication we have a proof that G is unprovable in T. (The proof is by simple modus ponens: If Con(T) then G; Con(T), therefore G). But this leads to a contradiction, because to have a proof that G is unprovable in T is just to have a proof of G, since G says "G is unprovable in T." The Second Incompleteness Theorem follows: If P is consistent (i.e. does not lead to contradictions) it is impossible to obtain a proof of the consistency of T.

The incompleteness theorems showed that no consistent axiomatic system was capable of capturing all the truths of arithmetic, and this result effectively put an end to nearly a half-century of efforts by mathematicians and logicians, beginning with Gottlob Frege, through Bertrand Russell, Alfred North Whitehead, and David Hilbert, to explain the foundations of mathematics solely in the terms of logic and set theory.

Some scholars have debated over what, if anything, Gödel's incompleteness theorems imply about anthropic mechanism. One of the earliest attempts to use incompleteness to reason about human intelligence was by Gödel himself in his 1951 Gibbs lecture entitled "Some basic theorems on the foundations of mathematics and their philosophical implications." In this lecture, Gödel uses the incompleteness theorem to arrive at the following disjunction: (a) the human mind is not a consistent finite machine, or (b) there exist Diophantine equations for which it cannot decide whether solutions exist. Gödel finds (b) implausible, and thus seems to have believed the human mind was not equivalent to a finite machine, i.e., its power exceeded that of any finite machine. He recognized that this was only a conjecture, since one could never disprove (b). Yet he considered the disjunctive conclusion to be a "certain fact." Other philosophers including J.R. Lucas, Hilary Putnam, Roger Penrose, and Douglas Hofstadter have discussed the potential implications of the incompleteness theorems in areas outside of the philosophy of mathematics. To learn more about the debate, see the entry on mechanism.

Set Theory

In 1935 Gödel proved that the Axiom of Choice was consistent with the other axioms of Zermelo-Fraenkel set theory (ZFC). Until then many mathematicians had been suspicious of the axiom because of some of its intuitively troubling implications. Gödel went on to prove in 1937 the consistency of Cantor's Continuum Hypothesis with ZFC, which led to his well-known 1940 paper, "The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory," in which he introduces the idea of a constructible universe. The constructible universe is a particular class of sets which can be described entirely in terms of simpler sets. Gödel proved that the constructible universe is an inner model of ZF set theory, and also that the Axiom of Choice and the Continuum Hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.

Legacy

The Kurt Gödel Society, founded in 1987, was named in his honor. It is an international organization for the promotion of research in the areas of logic, philosophy, and the history of mathematics.

Gödel's friendship with Einstein

Albert Einstein and Gödel had a legendary friendship, shared in the walks they took together to and from the Institute for Advanced Studies. The nature of their conversations was a mystery to the other Institute members. Economist Oskar Morgenstern recounts that toward the end of his life Einstein confided that his "own work no longer meant much, that he came to the Institute merely…to have the privilege of walking home with Gödel".[2]

Einstein and Morgenstern coached Gödel for his U.S. citizenship exam, concerned that their friend's unpredictable behavior might jeopardize his chances. When the Nazi regime was briefly mentioned, Gödel informed the presiding judge that he had discovered a way in which a dictatorship could be legally installed in the United States, through a logical contradiction in the U.S. Constitution. Neither judge, nor Einstein or Morgenstern allowed Gödel to finish his line of thought and he was awarded citizenship.[3]

Gödel in popular culture

In the 1994 romantic comedy "I.Q. (film)" directed by Fred Schepisi, Gödel was dramatized as a secondary character portrayed by actor Lou Jacobi. The film portrays Gödel without his paranoia and fully enjoying his retirement. In 2007 students from the Nederlandse Filmacademie (Dutch) (Dutch Film Academy) graduated with a 25-minute short "Gödel." It was directed by Igor Kramer with Austrian actor Robert Stuc in the title role. In this short a retired Gödel realizes his surroundings are a filmset, feeding his paranoia.

Notes

  1. ↑ Frederick Toates and Olga Coschug Toates. Obsessive Compulsive Disorder: Practical Tried-and-Tested Strategies to Overcome OCD. (Class Publishing, 2002. ISBN 978-1859590690), 221
  2. ↑ Rebecca Goldstein. Incompleteness: The Proof and Paradox of Kurt Godel. (New York: W. W. Norton, 2005. ISBN 978-0393051698), 33
  3. ↑ Jim Holt, "The Loophole: A logician challenges the Constitution." Lingua Franca (February 1998) [1]. accessdate November 17, 2007

Bibliography

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