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This 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. The topic should interest researchers and students who work on proof theory, history of proof theory or Hilbert's program and who do not mind reading mathematical texts.
List of contents
Acknowledgements.- 1 Introduction.- 2 Preliminaries.- 3 Ordinal numbers.- 4 Consistency proof.- Index.- References.
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From the book reviews:
“This book deals with G. Gentzen’s classical 1936 article on the consistency of arithmetic … . The result is a clear exposition of the full proof with all details and with the ideas behind the proof visible. … All in all, the author of this book has accomplished the difficult task of making Gentzen’s original hard-to-read paper accessible to the historically interested logician.” (Christian Bennet, Mathematical Reviews, November, 2014)
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From the book reviews:
"This book deals with G. Gentzen's classical 1936 article on the consistency of arithmetic ... . The result is a clear exposition of the full proof with all details and with the ideas behind the proof visible. ... All in all, the author of this book has accomplished the difficult task of making Gentzen's original hard-to-read paper accessible to the historically interested logician." (Christian Bennet, Mathematical Reviews, November, 2014)
