Gentzen's consistency proof is a result of proof theory in mathematical logic , published by Gerhard Gentzen in It shows that the Peano axioms of first-order arithmetic do not contain a contradiction i. Gentzen argued that it avoids the questionable modes of inference contained in Peano arithmetic and that its consistency is therefore less controversial. Gentzen's theorem is concerned with first-order arithmetic: the theory of the natural numbers , including their addition and multiplication, axiomatized by the first-order Peano axioms. This is a "first-order" theory: the quantifiers extend over natural numbers, but not over sets or functions of natural numbers.
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It seems that you're in Germany. We have a dedicated site for Germany. Gerhard Gentzen is best known for his development of the proof systems of natural deduction and sequent calculus, central in many areas of logic and computer science today. After these successes, Gentzen dedicated the rest of his short life to the main problem of Hilbert's proof theory, the question of the consistency of analysis.
He was arrested in the summer of with other professors of the German University of Prague and died soon afterward of starvation in a prison cell. Attempts at locating his lost manuscripts failed at the time, but several decades later, two slim folders of shorthand notes were found.
In this volume, Jan von Plato gives an overview of Gentzen's life and scientific achievements, based on detailed archival and systematic studies, and essential for placing the translations of shorthand manuscripts that follow in the right setting.
The materials in this book are singular in the way they show the birth and development of Gentzen's central ideas and results, sometimes in a well-developed form, and other times as flashes into the anatomy of the workings of a unique mind. The book is valuable source for the history of modern logic; the editor did an excellent work in getting the shorthand notes, first transcribed in normal German text, and then translating it to English.
Buy eBook. Buy Hardcover. Buy Softcover. FAQ Policy. About this book Gerhard Gentzen is best known for his development of the proof systems of natural deduction and sequent calculus, central in many areas of logic and computer science today.
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Gentzen's consistency proof
He made major contributions to the foundations of mathematics , proof theory , especially on natural deduction and sequent calculus. He died of starvation in a Soviet prison camp in Prague in , having been interned as a German national after the Second World War. Bernays was fired as "non- Aryan " in April and therefore Hermann Weyl formally acted as his supervisor. Gentzen joined the Sturmabteilung in November although he was by no means compelled to do so.
Eckart Menzler-Trott. American Mathematical Soc. Gerhard Gentzen is the founder of modern structural proof theory. His lasting methods, rules, and structures resulted not only in the technical mathematical discipline called ''proof theory'' but also in verification programs that are essential in computer science. The appearance, clarity, and elegance of Gentzen's work on natural deduction, the sequent calculus, and ordinal proof theory continue to be impressive even today.