Saved from the Cellar : Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics Gerhard Gentzen is best known for his development of the proof systems of natural deduction and sequent calculus central in many areas of logic and computer science today. Another noteworthy achievement is his resolution of the embarrassing situation created by Gödels incompleteness results especially the second one about the unprovability of consistency of elementary arithmetic. After these successes Gentzen dedicated the rest of his short life to the main problem of Hilberts proof theory the question of the consistency of analysis. He was arrested in the summer of 1945 with other professors of the German University of Prague and died soon afterward of starvation in a prison cell. Attempts at locating his lost manuscripts failed at the time but several decades later two slim folders of shorthand notes were found. In this volume Jan von Plato gives an overview of Gentzens life and scientific achievements based on detailed archival and systematic studies and essential for placing the translations of shorthand manuscripts that follow in the right setting. The materials in this book are singular in the way they show the birth and development of Gentzens central ideas and results sometimes in a well developed form and other times as flashes into the anatomy of the workings of a unique mind Part I: A Sketch of Gentzens Life and Work 1. Overture 2. Gentzens years of study Dr. Gentzens arduous years in Nazi Germany 4. The scientific accomplishments 5. Loose ends 6. Gentzens genuis Part II: Overview of the Shorthand Notes 1. Gentzens series of stenographic manuscripts 2. The items in this collection Practical remarks on the manuscripts Manuscript illustrations The German alphabet in Latin Sutterlin and Fraktur Type Bibliography for parts I and II Index of names for Parts I and II Part III: The Original Writings 1. Reduction of number theoretic problems to predicate logic 2. Replacement of functions by predicates 3. The formation of abstract concepts 4. Five different forms of natural calculi 5. Formal conception of correctness in arithmetic I 6. Investigations into logical inferences 7. Reduction of classical to intuitionistic logic 8. CV of the candidate Gerhard Gentzen. 0 9. Letters to Heyting 10. Formal conception of correctness in arithmetic II 11. Proof theory of number theory 12. Consistency of artihmetic for publication 13. Correspondence with Paul Bernays 14. Forms of type theory 15. Predicate logic 16. Propositional logic 17. Foundational research in mathematics Table of cross references in the Gentzen papers Index of names in the Gentzen papers Index of subjects in the Gentzen papers