Editing Gerhard Gentzen (section)

==Work==
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]].

One of Gentzen's papers had a second publication in the ideological ''[[Deutsche Mathematik]]'' that was founded by [[Ludwig Bieberbach]] who promoted "Aryan" mathematics.{{sfn|Tydecks|2002}}

Gentzen [[Gentzen's consistency proof|proved the consistency]] of the [[Peano axioms]] in a paper published in 1936.{{sfn|Gentzen|1936b}} In his [[Habilitation|''Habilitationsschrift'']], 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]].