Where is the Gödel-Point hiding: Gentzen's Consistency Proof of 1936 and his Representation of Constructive Ordinals

Abstract

The book explains the first published consistency proof of PA. It contains the original Gentzen's proof, but it uses modern terminology and examples to illustrate the essential notions.

The author comments on Gentzen's steps which are supplemented with exact calculations and parts of formal derivations. A notable aspect of the proof is the representation of ordinal numbers that was developed by Gentzen.

This representation is analysed and connection to set-theoretical representation is found, namely an algorithm for translating Gentzen's notation into Cantor normal form.

Keywords

• Algorithm for translating Gentzen's notation of ordinal numbers
• Gentzen's notation and standard notation of ordinal numbers
• Gerhard Gentzen
• Hilbert's program
• Ordinal numbers up to ε_0
• Peano Arithmetic
• Transfinite induction up to ε_0