Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, cosmologist, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century (at a time when Bertrand Russell,For instance, in their "Principia Mathematica (Stanford Encyclopedia of Philosophy edition). Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building on earlier work by Frege, Richard Dedekind, and Georg Cantor.
Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two years later, in 1931. The incompleteness theorems address limitations of formal axiomatic systems. In particular, they imply that a formal axiomatic system satisfying certain technical conditions cannot decide the truth value of all statements about the natural numbers, and cannot prove that it is itself consistent.Smullyan, R. M. (1992). Gödel's Incompleteness Theorems. New York, Oxford: Oxford University Press, ch. V.Smullyan, R. M. (1992). Gödel's Incompleteness Theorems. New York, Oxford: Oxford University Press, ch. IX. To prove this, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
Gödel also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo–Fraenkel set theory, assuming that its axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
Born into a wealthy German-speaking family in Brno, Gödel emigrated to the United States in 1939 to escape the rise of Nazi Germany. Later in life, he suffered from mental illness; believing that his food was being poisoned, he refused to eat and starved to death.
Early life and education Childhood Gödel was born on April 28, 1906, in Brünn, Austria-Hungary (now Brno, Czech Republic), into the German-speaking family of Rudolf Gödel, the managing director and part owner of a major textile firm, and Marianne Gödel (née Handschuh). His father was Catholic and his mother was Protestant, and the children were raised as Protestants. Many of Kurt Gödel's ancestors were active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer in his time and for some years a member of the (Men's Choral Union of Brünn).Procházka 2008, pp. 30–34.
Gödel automatically became a citizen of Czechoslovakia at age 12 when the Austro-Hungarian Empire collapsed following its defeat in the First World War. According to his classmate , like many residents of the predominantly German , "Gödel considered himself always Austrian and an exile in Czechoslovakia".Dawson 1997, p. 15. In February 1929, he was granted release from his Czechoslovak citizenship and then, in April, granted Austrian citizenship. When Germany annexed Austria in 1938, Gödel automatically became a German citizen at age 32. In 1948, after World War II, at age 42, he became a U.S. citizen.
In his family, the young Gödel was nicknamed ("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 remained convinced for the rest of his life that his heart had been permanently damaged. Beginning at age four, Gödel had "frequent episodes of poor health", which continued all his life.
Gödel attended the , a Lutheran school in Brünn, from 1912 to 1916, and was enrolled in the from 1916 to 1924, excelling with honors in all subjects, particularly mathematics, languages, and religion. Although he had first excelled in languages, he became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf left for Vienna, where he attended medical school at the University of Vienna. During his teens, Gödel studied Gabelsberger shorthand, criticism of Isaac Newton, and the writings of Immanuel Kant.
Studies in Vienna
At age 18, Gödel joined his brother at the University of Vienna. He had already mastered university-level mathematics.Dawson 1997, p. 24. Although initially intending to study theoretical physics, he also attended courses on mathematics and philosophy.At the University of Vienna, Gödel attended mathematics and philosophy courses side by side with Hermann Broch, who was in his early forties. See: During this time, he adopted ideas of mathematical realism. He read Kant's , and participated in the Vienna Circle with Moritz Schlick, Hans Hahn, and Rudolf Carnap. Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick that studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."Gleick, J. (2011) The Information: A History, a Theory, a Flood, London, Fourth Estate, p. 181.
Attending a lecture by David Hilbert in Bologna on completeness and consistency in mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published (Principles of Mathematical 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?"
Gödel chose this topic for his doctoral work. In 1929, aged 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established his eponymous completeness theorem regarding first-order logic. He was awarded his doctorate in 1930, and his thesis (accompanied by additional work) was published by the Vienna Academy of Science.
In 1929 Gödel met (née Porkert), a divorcee living with her parents across the street from him.Dawson Jr., John W., and Karl Sigmund. “Gödel’s Vienna.” Mathematical Intelligencer, vol. 28, no. 3, Summer 2006, Page 46. EBSCOhost, https://doi.org/10.1007/BF02986884.M The two married (in a civil ceremony) a decade later, in September 1938. A trained ballet dancer, Adele was working as a masseuse at the time they met. At one point she worked as a dancer at a downtown nightclub called the Nachtfalter ("nocturnal moth"). Gödel's parents opposed their relationship because of her background and age (six years older than he).Wang 1987. Page 80. It appears to have been a happy marriage.Brewer, William D. "Kurt Gödel: The Genius of Metamathematics". Springer Nature. 2022. Page 250 Adele was an important support to Gödel, whose psychological problems affected their daily lives.Toates, Frederick. Olga Coschug-Toates. "Obsessive Compulsive Disorder: Practical, Tried-and-tested Strategies to Overcome OCD." Class Publishing Ltd. 2002. Page 221. The two had no children.
Career
Incompleteness theorems
In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg on September 5–7. There, he presented his completeness theorem of first-order logic, and, at the end of the talk, mentioned that this result does not generalise to higher-order logic, thus hinting at his incompleteness theorems.
Gödel published his incompleteness theorems in (called in English "On Formally Undecidable Propositions of Principia Mathematica and Related Systems"). In that article, he proved for any computable axiomatic system powerful enough to describe the arithmetic of the natural numbers (e.g., the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:
If a (logical or axiomatic formal) system is omega-consistent, it cannot be syntactically complete. The consistency of axioms cannot be proved within their own system. These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in and Hilbert's program, to find a non-relatively consistent axiomatization sufficient for number theory (that was to serve as the foundation for other fields of mathematics).
Gödel constructed a formula that claims it is itself unprovable in a given formal system. If it were provable, it would be false. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that is true of arithmetic, but not provable in that system. To make this precise, Gödel had to produce a method to encode (as natural numbers) statements, proofs, and the concept of provability; he did this by a process known as Gödel numbering.
In his two-page paper (1932), Gödel refuted the finite-valuedness of intuitionistic logic. In the proof, he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).
Mid-1930s: further work and U.S. visits Gödel earned his habilitation at Vienna in 1932, and in 1933 became a (unpaid lecturer) there. In 1933, Adolf Hitler came to power in Germany, and over the following years the Nazis rose in influence in Austria and among Vienna's mathematicians. In June 1936, Moritz Schlick, whose seminar had aroused Gödel's interest in logic, was murdered by one of his former students, Johann Nelböck. This triggered "a severe nervous crisis" in Gödel.. From p. 80, which quotes Rudolf Gödel, Kurt's brother and a medical doctor. The words "a severe nervous crisis", and the judgment that Schlick's murder was its trigger, are Rudolf Gödel's. Rudolf knew Kurt well in those years. He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.Dawson 1997, pp. 110–12
In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend.Hutchinson Encyclopedia (1988), p. 518 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 that he was able to present 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, titled On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his PhD at Princeton, took notes on these lectures that were later published.
Gödel visited the IAS again in the autumn of 1935. The traveling and hard work had exhausted him and the next year he took a break to recover from a depressive episode. 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 went on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.