Gerhard Gentzen

Gerhard Gentzen (November 24th 1909 with Greifswald - August 4th 1945 with Prague) was a Mathématicien and German Logicien . Its work is fundamental in Théorie of the demonstration.

It was one of the students of Weyl to the Université of Göttingen of 1929 with 1933. He invented two systems of deduction for first order logic, the natural Déduction and the Calcul of the séquents. For this last, it showed its Hauptsatz (principal theorem), published in 1934 in its Recherches on the logical deduction .

the fundamental theorem affirms that any purely logical demonstration can be brought back to a given normal form, which is by no means univocal besides. One can formulate the essential properties of such a normal demonstration about in the following way: it does not comprise turnings. One introduced there no concept which is not contained in its end result and which, consequently, should not necessarily be used to obtain this result. ( Research on the logical deduction , seen overall, p.4-5)

Dag Prawitz showed in 1965 a similar theorem for the natural deduction.

Gentzen is famous to have shown the coherence of arithmetic of Peano (in 1936) by using a principle of induction until ordinal countable the ε₀, but for formulas of low logical complexity. The methods used for this proof proved to be essential for the Théorie of the modern demonstration. The theory in which this proof can be formalized is necessarily stronger than the arithmetic one of Peano according to the second Théorème of incomplétude of Gödel (with the direction where if it makes it possible to prove the coherence of arithmetic of Peano, its coherence will not be able thus to be proven in this arithmetic). One could see this proof, in which Gödel was interested much, like an attempt to rehabilitate the program of Hilbert, by widening the concept of methods finitaires to inductions until certain ordinal like ε₀. The coherence of the theory used by Gentzen for its proof, although stronger, would be less doubtful than the coherence of arithmetic of Peano, because induction, although until ordinal (inevitably higher than that of the entireties), is done on simple formulas. That is hardly constant such as it is. In a more objective way, this proof makes it possible to analyze the coherence of arithmetic of Peano; for example the result of coherence makes it possible to measure the " of it; force" by ordinal the ε₀. While generalizing, one thus could engage a classification of the arithmetic theories.

Gentzen died in a prison camp of war, after being stopped by the Soviets because of its honesties Nazis.

See too

Theorem of incomplétude of Gödel
fundamental articles of Gentzen on the Web (in German): article 1 and article 2

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