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.

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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. The theory is strong enough to describe recursively defined integer functions such as exponentiation, factorials or the Fibonacci sequence. Primitive recursive arithmetic is a much simplified form of arithmetic that is rather uncontroversial. The additional principle means, informally, that there is a well-ordering on the set of finite rooted trees.

It is the limit of the sequence:. To express ordinals in the language of arithmetic, an ordinal notation is needed, i. This can be done in various ways, one example provided by Cantor's normal form theorem. Gentzen defines a notion of "reduction procedure" for proofs in Peano arithmetic. For a given proof, such a procedure produces a tree of proofs, with the given one serving as the root of the tree, and the other proofs being, in a sense, "simpler" than the given one.

It is possible to interpret Gentzen's proof in game-theoretic terms Tait It is sometimes claimed that the consistency of a theory can only be proved in a stronger theory.

Gentzen's theory obtained by adding quantifier-free transfinite induction to primitive recursive arithmetic proves the consistency of first-order Peano arithmetic PA but does not contain PA.

For example, it does not prove ordinary mathematical induction for all formulae, whereas PA does since all instances of induction are axioms of PA. Gentzen's theory is not contained in PA, either, however, since it can prove a number-theoretical fact—the consistency of PA—that PA cannot.

Therefore, the two theories are, in one sense, incomparable. That said, there are other, more powerful ways to compare the strength of theories, the most important of which is defined in terms of the notion of interpretability. It can be shown that, if one theory T is interpretable in another B, then T is consistent if B is. Indeed, this is a large point of the notion of interpretability. And, assuming that T is not extremely weak, T itself will be able to prove this very conditional: If B is consistent, then so is T.

Hence, T cannot prove that B is consistent, by the second incompleteness theorem, whereas B may well be able to prove that T is consistent. This is what motivates the idea of using interpretability to compare theories, i. So, in the sense of consistency strength, as characterized by interpretability, Gentzen's theory is stronger than Peano arithmetic. Kleene , p. Gentzen's first version of his consistency proof was not published during his lifetime because Paul Bernays had objected to a method implicitly used in the proof.

The modified proof, described above, was published in in the Annals. Gentzen went on to publish two more consistency proofs, one in and one in Gentzen's proof is the first example of what is called proof-theoretical ordinal analysis. In ordinal analysis one gauges the strength of theories by measuring how large the constructive ordinals are that can be proven to be well-ordered, or equivalently for how large a constructive ordinal can transfinite induction be proven.

A constructive ordinal is the order type of a recursive well-ordering of natural numbers. Laurence Kirby and Jeff Paris proved in that Goodstein's theorem cannot be proven in Peano arithmetic. Their proof was based on Gentzen's theorem.

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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.

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## Gerhard Gentzen

View one larger picture. It was there that Gerhard spent his childhood years, attending first the elementary school there, and later the Realgymnasium. 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 it was with distinction and he was ranked top in his school.

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## Saved from the Cellar

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.