Theory of the demonstration

The theory of the demonstration , so known under the name of theory of the proof , is a branch of the Logique mathematics. It was founded by David Hilbert at the beginning of the 20th century.

Hilbert proposed this new mathematical discipline at the time of sound celebrates exposed with the 2nd international Congrès of mathematics in 1900 with for objective solving the problem of the coherence of mathematics. This objective was invalidated by the not less famous Théorème of incomplétude of Gödel in 1931, which however did not prevent the theory of the demonstration from developing, in particular thanks to work of Jacques Herbrand and Gerhard Gentzen. This last showed one of the principal results of the theory of the demonstration, known under the name of Hauptsatz (principal theorem) or theorem of elimination of the cuts. Gentzen then used this theorem to give the first purely syntactic proof of the coherence of the Arithmétique.

After one period of calm, which all the same made it possible to determine a certain number of other results of relative coherence and to outline a classification of the axiomatic theories, the theory of the demonstration knew a spectacular rebirth during the years 1960 with the discovery of the Correspondance of Curry-Howard which exhibé a structural bond new and major between logic and informatique : primarily the procedure of elimination of the cuts defined by Gentzen can be seen like a computing process, so that the formal demonstrations become programs then. Since, the theory of the demonstration developed in narrow symbiosis with other fields of logic and theoretical data processing, in particular the Lambda-calculation, and gave rise to new models of the calculation, most recent being the linear Logique of Jean-Yves Girard in 1986. Today, part of the theory of the demonstration merges with the Sémantique computer programming languages and interacts with many other disciplines of logic or data processing théorique :

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