Kurt Gödel

Kurt Friedrich Gödel (/ˈɡɜːrdəl/ GUR-dəl,[2] German: [kʊʁt ˈɡøːdl̩] ^ⓘ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell,[3] Alfred North Whitehead,[3] and David Hilbert were using logic and set theory to investigate the foundations of mathematics, building on earlier work by the likes of Richard Dedekind, Georg Cantor and Gottlob Frege.

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Kurt Gödel
Gödel c. 1926
Born	Kurt Friedrich Gödel (1906-04-28)April 28, 1906 Brünn, Austria-Hungary (now Brno, Czech Republic)
Died	January 14, 1978(1978-01-14) (aged 71) Princeton, New Jersey, U.S.
Citizenship	Austria Czechoslovakia Germany United States
Alma mater	University of Vienna (PhD, 1930)
Known for	Gödel's incompleteness theorems Gödel's completeness theorem Gödel's constructible universe Gödel metric (closed timelike curve) Gödel logic Gödel–Dummett logic Gödel's β function Gödel numbering Gödel operation Gödel's speed-up theorem Gödel's ontological proof Gödel–Gentzen translation Gödel–McKinsey–Tarski translation Von Neumann–Bernays–Gödel set theory ω-consistent theory The consistency of the continuum hypothesis with ZFC Axiom of constructibility Compactness theorem Condensation lemma Diagonal lemma Dialectica interpretation Ordinal definable set Slingshot argument
Spouse	Adele Nimbursky ​ (m. 1938)​
Awards	Albert Einstein Award (1951) ForMemRS (1968)[1] National Medal of Science (1974)
Scientific career
Fields	Mathematics, mathematical logic, analytic philosophy, physics
Institutions	Institute for Advanced Study
Thesis	Über die Vollständigkeit des Logikkalküls (1929)
Doctoral advisor	Hans Hahn
Signature

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 first incompleteness theorem states that for any ω-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example, Peano arithmetic), there are true propositions about the natural numbers that can be neither proved nor disproved from the axioms.[4] To prove this, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers. The second incompleteness theorem, which follows from the first, states that the system cannot prove its own consistency.[5]

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.