# Gödel and Intuitionism

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## Abstract

After a brief survey of Gödel’s personal contacts with Brouwer and Heyting, examples are discussed where intuitionistic ideas had a direct influence on Gödel’s technical work. Then it is argued that the closest rapprochement of Gödel to intuitionism is seen in the development of the Dialectica Interpretation, during which he came to accept the notion of computable functional of finite type as primitive. It is shown that Gödel already thought of that possibility in the Princeton lectures on intuitionism of Spring 1941, and evidence is presented that he adopted it in the same year or the next, long before the publication of 1958. Draft material for the revision of the Dialectica paper is discussed in which Gödel describes the Dialectica Interpretation as being based on a new intuitionistic insight obtained by applying phenomenology, and also notes that relate the new notion of reductive proof to phenomenology. In an appendix, attention is drawn to notes from the archive according to which Gödel anticipated autonomous transfinite progressions when writing his incompleteness paper.

## Keywords

Finite Type Computable Function Intuitionistic Logic Original Emphasis Incompleteness Theorem## Notes

### Acknowledgements

This is the revised and much extended text of the talk with the same title given at the conference ‘Calculability and constructivity: historical and philosophical aspects’ of the International Union of the History and Philosophy of Science (Joint Session of the Division of Logic, Methodology and Philosophy of Science and of the Division of the History of Science and Technology), Paris, November 18, 2006. Much of that talk was derived from a manuscript that has in the meantime appeared as part of the present author’s contribution to van Atten and Kennedy (2009) (written in 2005). Other versions of that talk were presented at the plenary discussion ‘Gödel’s Legacy’ at the ASL European Summer Meeting in Nijmegen, August 2, 2006 and at seminars in Nancy (2005), Tokyo (2006), Utrecht (2006), and Aix-en-Provence (2007). I am grateful to the respective organisers for the invitations, and to the audiences for their questions, criticisms, and comments.

The quotations from Gödel’s notebooks and lecture notes appear courtesy of the Kurt Gödel Papers, The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, USA, on deposit at Princeton University. I am grateful to Marcia Tucker, Christine Di Bella, and Erica Mosner of the Historical Studies-Social Science Library at the IAS for their assistance in finding anwers to various questions around this material. In the study of Gödel’s notes in Gabelsberger shorthand, I have been able to consult Cheryl Dawson’s transcriptions, which she generously made available to me; these were also useful to Robin Rollinger and Eva-Maria Engelen, to whom I am greatly indebted for additional, speedy help with the shorthand, also concerning previously untranscribed passages. Access to the microfilm edition of the Kurt Gödel Papers was kindly provided to Rollinger, Engelen and me by Gabriella Crocco. The present paper is realized as part of her project ‘Kurt Gödel philosophe: de la logique à la cosmologie’, funded by the Agence Nationale de Recherche (project number BLAN-NT09-436673), whose support is gratefully acknowledged.

Gödel’s letters to his brother quoted here are part of a collection of letters that was found in 2006. I am grateful to Matthias Baaz and Karl Sigmund for bringing this correspondence to my attention, and for providing me with photocopies. These letters have been deposited at the Wienbibliothek im Rathaus, Vienna. The quotations appear courtesy of the Kurt Gödel Papers, The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, USA.

I am grateful to Dirk van Dalen, Georg Kreisel, Albert Visser, and, in particular, William Howard and Göran Sundholm, for comments, references, criticisms and discussion. An anonymous referee wrote a helpful report on an earlier version, and Jaime Gaspar helpfully sent a list of typing errors.

Prof. Howard kindly granted permission to quote from the reminiscences he sent me; some of these come from the notes he prepared for Amy Shell-Gellasch, who used them for her article Shell-Gellasch (2003). Those notes are now held at the Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin, as part of the William Howard Oral History Collection, 1973, 1990–2003. These Archives hold the copyright; quotations are by permission. I thank its staff member Carol Mead for her help and advice concerning this material and its use.

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