Proof theory
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| - | '''Gerhard Karl Erich Gentzen''' (November 24, 1909 – August 4, 1945) was a [[Germany|German]] [[mathematician]] and [[logician]]. He made major contributions to the [[foundations of mathematics]], [[proof theory]], especially on [[natural deduction]] and [[sequent calculus]]. He died in 1945 after the [[Second World War]], because he was deprived of food after being arrested in [[Prague]]. | + | '''Proof theory''' is a branch of [[mathematical logic]] that represents [[Mathematical proof|proof]]s as formal [[mathematical object]]s, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined [[data structures]] such as plain lists, boxed lists, or trees, which are constructed according to the [[axiom]]s and [[rule of inference|rules of inference]] of the logical system. As such, proof theory is [[syntax (logic)|syntactic]] in nature, in contrast to [[model theory]], which is [[Formal semantics (logic)|semantic]] in nature. Together with [[model theory]], [[axiomatic set theory]], and [[recursion theory]], proof theory is one of the so-called ''four pillars'' of the [[foundations of mathematics]]. |
| - | ==Life and career== | + | ==See also== |
| - | Gentzen was a student of [[Paul Bernays]] at the [[University of Göttingen]]. Bernays was fired as "non-[[Aryan]]" in April 1933 and therefore [[Hermann Weyl]] formally acted as his supervisor. Gentzen joined the [[Sturmabteilung]] in November 1933 although he was by no means compelled to do so. Nevertheless he kept in contact with Bernays until the beginning of the [[Second World War]]. In 1935, he corresponded with [[Abraham Fraenkel]] in Jerusalem and was implicated by the Nazi teachers' union as one who "keeps contacts to the [[Chosen People]]." In 1935 and 1936, [[Hermann Weyl]], head of the Göttingen mathematics department in 1933 until his resignation under Nazi pressure, made strong efforts to bring him to the [[Institute for Advanced Study]] in Princeton. | + | *[[Intermediate logic]] |
| + | *[[Model theory]] | ||
| + | *[[Proof (truth)]] | ||
| + | *[[Proof techniques]] | ||
| - | Between November 1935 and 1939 he was an assistant of [[David Hilbert]] in Göttingen. Gentzen joined the [[Nazi Party]] in 1937. In April 1939 Gentzen swore the oath of loyalty to [[Adolf Hitler]] as part of his academic appointment. From 1943 he was a teacher at the [[Karl-Ferdinands-Universität|University of Prague]]. Under a contract from the [[Schutzstaffel|SS]] Gentzen evidently worked for the [[V-2]] project. | ||
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| - | After the war he starved to death in [[Prague]], after being arrested like all other Germans in Prague on May 7, 1945 and deprived of food. | ||
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| - | ==Work== | ||
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| - | Gentzen's main work was on the foundations of [[mathematics]], in [[proof theory]], specifically [[natural deduction]] and the [[sequent calculus]]. His [[cut-elimination theorem]] is the cornerstone of [[proof-theoretic semantics]], and some philosophical remarks in his "Investigations into Logical Deduction", together with [[Ludwig Wittgenstein]]'s later work, constitute the starting point for [[inferential role semantics]]. | ||
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| - | One of Gentzen's papers had a second publication in the ideological ''[[Deutsche Mathematik]]'' that was founded by [[Ludwig Bieberbach]] who promoted "Aryan" mathematics. | ||
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| - | Gentzen [[Gentzen's consistency proof|proved the consistency]] of the [[Peano axioms]] in a paper published in 1936. In his [[Habilitation]]sschrift, finished in 1939, he determined the proof-theoretical strength of Peano arithmetic. This was done by a direct proof of the unprovability of the principle of transfinite induction, used in his 1936 proof of consistency, within Peano arithmetic. The principle can, however, be expressed in arithmetic, so that a direct proof of [[Gödel's incompleteness theorem]] followed. [[Kurt Gödel|Gödel]] used a coding procedure to construct an unprovable formula of arithmetic. Gentzen's proof was published in 1943 and marked the beginning of [[Ordinal analysis|ordinal proof theory]]. | ||
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Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Together with model theory, axiomatic set theory, and recursion theory, proof theory is one of the so-called four pillars of the foundations of mathematics.
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