Decoding Gentzen's Notation

Bellotti, Luca

doi:10.1080/01445340.2017.1422092

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γ → ϕ(0)1

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2019

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“…Gentzen's original consistency proof and his ordinal notation are explained in detail in [25]. An even more recent presentation of Gentzen's proof and its interpretation in the ordinary set-theoretic notation can be found in [5].…”

Section: γ → ϕ(0)mentioning

confidence: 99%

Nominalistic Ordinals, Recursion on Higher Types, and Finitism

Hämeen-Anttila¹

2019

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In 1936, Gerhard Gentzen published a proof of consistency for Peano Arithmetic using transfinite induction up to ε0, which was considered a finitistically acceptable procedure by both Gentzen and Paul Bernays. Gentzen’s method of arithmetising ordinals and thus avoiding the Platonistic metaphysics of set theory traces back to the 1920s, when Bernays and David Hilbert used the method for an attempted proof of the Continuum Hypothesis. The idea that recursion on higher types could be used to simulate the limit-building in transfinite recursion seems to originate from Bernays. The main difficulty, which was already discovered in Gabriel Sudan’s nearly forgotten paper of 1927, was that measuring transfinite ordinals requires stronger methods than representing them. This paper presents a historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays’ finitary interpretation of transfinite induction.

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Section: γ → ϕ(0)mentioning

confidence: 99%

Nominalistic Ordinals, Recursion on Higher Types, and Finitism

Hämeen-Anttila¹

2019

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show abstract

Decoding Gentzen's Notation

Cited by 1 publication

References 3 publications

Nominalistic Ordinals, Recursion on Higher Types, and Finitism

Nominalistic Ordinals, Recursion on Higher Types, and Finitism

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