**Gerhard Gentzen** invented a 'natural deduction' which provided a logic closer to mathematical reasoning than the systems proposed by Frege, Russell and Hilbert.

- Gentzen had already begun his secondary schooling at this stage but he continued his education at the Humanistische Gymnasium in Stralsund.
- Certainly moving schools did not affect Gentzen's academic achievements for when he received his Abitur in 1928 it was with distinction and he was ranked top in his school.
- Gentzen, as was usual at this time, moved between different German universities.
- In 1933 Gentzen was awarded his doctorate by Göttingen but the intense study in different environments had taken its toll so he was forced at this stage to return home to rest and recover his health.
- During these years Gentzen published some of his most important papers and was also given the responsible task of reviewing numerous works of eminent researchers from many countries for the Zentralblatt für Mathematik.
- As we have mentioned, Gentzen's work was on logic and the foundations of mathematics.
- This idea was later attributed to Tarski who introduced it in 1936, three years after Gentzen.
- In 1934 Gentzen gave the method of succinct Sequenzen, rules of consequents, which were particularly useful for deriving metalogical decidability results.
- Gentzen wrote several papers on these concepts, particularly examining the occurrence of set theory paradoxes.
- Of course Gödel published his incompleteness theorem just at the time Gentzen was beginning his work.
- At first Gentzen worried that it affected what he wanted to achieve on the foundations of mathematics and he withdrew what would have been his second paper after he had corrected the final proofs because of worries about the significance of Gödel's theorems.
- In a paper published in Mathematische Zeitschrift in 1935 Gentzen introduced two new versions of predicate logic now called the NNN-system and the LLL-system.
- By Gödel's unprovability theorem, such a proof as Gentzen gave had to make use of tools stronger than those of S; extending ordinary mathematical induction, Gentzen employed transfinite induction up to Cantor's first epsilon number, and he also showed that this was the minimum required for such proof.
- to what extent the Gentzen proof can be accepted as securing classical number theory in the sense of that problem formulation is in the present state of affairs a matter of individual judgement.
- Gentzen's was the most outstanding contribution to Hilbert's programme of axiomatising mathematics.
- Gentzen remained on the staff at Göttingen until 1943, although he had to undertake military service in the years 1939 until 1941.
- Gentzen was interned by the Russian forces and held in poor conditions.

Born 24 November 1909, Greifswald, Germany. Died 4 August 1945, Prague, Czechoslovakia.

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Analysis, Origin Germany

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive