Metatheorems | Proof theory

Gentzen's consistency proof

Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradiction (i.e. are "consistent"), as long as a certain other system used in the proof does not contain any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0", is neither weaker nor stronger than the system of Peano axioms. Gentzen argued that it avoids the questionable modes of inference contained in Peano arithmetic and that its consistency is therefore less controversial. (Wikipedia).

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From playlist Proofs

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From playlist Proofs

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Related pages

Fibonacci number | Primitive recursive arithmetic | Solomon Feferman | Ordinal analysis | Interpretability | Wilhelm Ackermann | Transfinite induction | Factorial | Peano axioms | Paul Bernays | Ordinal number | Tree (set theory) | Robinson arithmetic | Proof theory | Large countable ordinal | Ordinal arithmetic | Natural number | Quantifier (logic) | Hermann Weyl | Mathematical logic | Gödel's completeness theorem | Ordinal notation | Goodstein's theorem | Comparability | Recursion