Kurt Godel: Results on Foundations

Kurt Gödel (1906-1978) gained world-wide fame by his incompleteness theorem of 1931. Later, he set as his aim to solve what are known as Hilbert’s first and second problems, namely Cantor’s continuum hypothesis about the cardinality of real numbers, and secondly the consistency of the theory of real numbers and functions. By 1940, he was halfway through the first problem, in what was his last published result in logic and foundations. His intense attempts thereafter at solving these two problems have remained behind the veil of a forgotten German shorthand he used in all of his writing.

Results on Foundations is a set of four shorthand notebooks written in 1940-42 that collect results Gödel considered finished. Its main topic is set theory in which Gödel anticipated several decades of development. Secondly, Gödel completed his 1933 program of establishing the connections between intuitionistic and modal logic, by methods and results that today are at the same time new and 80 years old.

The present edition of Gödel’s four notebooks encompasses the 368 numbered pages and 126 numbered theorems of the Results on Foundations, together with a list of 74 problems on set theory Gödel prepared in 1946, and a list of an unknown date titled “The grand program of my research in ca. hundred questions.”

Table of Contents

PART I: GODEL’S RESULTS ON FOUNDATIONS

1. What is Resultate Grundlagen?

2. Main themes of Resultate Grundlagen

3. Description of the notebooks and editorial remarks

4. Godel’s lists of problems, results, questions, and remarks

5. And next?

6. Contents and short descriptions of Resultate Grundlagen

7. Godel’s own list of items in Resultate Grundlagen

PART II: Resultate Grundlagen

Notebook I

Notebook II

Notebook III

Notebook IV

PART III: ADDITIONAL ITEMS AND PROBLEM LISTS

Decision procedure for the positive propositional calculus

Russell’s antinomy with positive propositional calculus

Two additional notes

Questions and remarks on foundations

Problems and results 1–74, 1946

Grand program of my research

About the Authors

​Maria Hämeen-Anttila is a post-doctoral researcher at the University of Helsinki, Finland. Her previous work includes an edition of Gödel’s – The Princeton Lectures on Intuitionism, 2021.

Jan von Plato is a Research Director at the University of Helsinki. He has published nine books and numerous articles on foundations of probability, constructive axiomatics, proof theory, and the development of logic and foundational study.