Understanding the works of Gerhard Gentzen.

What are Gerhard Gentzen's mathematical accomplishments?

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...tzen, like the intuitionists, attributes the antimonies to a liberal use of infinity, but does not completely reject the law of the excluded middle in infinite situations. Gentzen showed in paper #2 [1934] that propositions of elementary number theory (the "first level" of mathematics) not involving any disjunction or existential quantifier is intuitionistically provable if and only if it is classically provable. He also proves that classical arithmetic can be interpreted in intuitionistic arithmetic using the same "double negation" transformation used by Kolmogorov [1925]. Gentzen notes that this implies, due to Godel's second incompleteness theorem, that ...