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Kurt Godel and the Foundations of Mathematics

Book Description

This volume commemorates the life, work and foundational views of Kurt Gödel (1906–1978), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency – with the other widely accepted axioms of set theory – of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Dedication
  5. Contents
  6. Contributors
  7. Foreword – Gaisi Takeuti
  8. Preface
  9. Acknowledgments
  10. Short Biography of Kurt Gödel
  11. Part I: Historical Context: Gödel’s Contributions and Accomplishments
    1. Gödel’s Historical, Philosophical, and Scientific Work
      1. Chapter 1: The Impact of Gödel’s Incompleteness Theorems on Mathematics
      2. Chapter 2: Logical Hygiene, Foundations, and Abstractions: Diversity among Aspects and Options
    2. Gödel’s Legacy: A Historical Perspective
      1. Chapter 3: The Reception of Gödel’s 1931 Incompletability Theorems by Mathematicians, and Some Logicians, to the Early 1960s
      2. Chapter 4: “Dozent Gödel Will Not Lecture”
      3. Chapter 5: Gödel’s Thesis: An Appreciation
      4. Chapter 6: Lieber Herr Bernays! Lieber Herr Gödel! Gödel on Finitism, Constructivity, and Hilbert’s Program
    3. The Past and Future of Computation
      1. Chapter 7: Computation and Intractability: Echoes of Kurt Gödel
      2. Chapter 8: From the Entscheidungsproblem to the Personal Computer – and Beyond
    4. Gödelian Cosmology
      1. Chapter 9: Gödel, Einstein, Mach, Gamow, and Lanczos: Gödel’s Remarkable Excursion into Cosmology
      2. Chapter 10: Physical Unknowables
  12. Part II: A Wider Vision: The Interdisciplinary, Philosophical, and Theological Implications of Gödel’s Work
    1. On the Unknowables
      1. Chapter 11: Gödel and Physics
      2. Chapter 12: Gödel, Thomas Aquinas, and the Unknowability of God
    2. Gödel and the Mathematics of Philosophy
      1. Chapter 13: Gödel’s Mathematics of Philosophy
    3. Gödel and Philosophical Theology
      1. Chapter 14: Gödel’s Ontological Proof and Its Variants
    4. Gödel and the Human Mind
      1. Chapter 15: The Gödel Theorem and Human Nature
      2. Chapter 16: Gödel, the Mind, and the Laws of Physics
  13. Part III: New Frontiers: Beyond Gödel’s Work in Mathematics and Symbolic Logic
    1. Extending Gödel’s Work
      1. Chapter 17: Gödel’s Functional Interpretation and Its Use in Current Mathematics
      2. Chapter 18: My Forty Years on His Shoulders
    2. The Realm of Set Theory
      1. Chapter 19: My Interaction with Kurt Gödel: The Man and His Work
    3. Gödel and the Higher Infinite
      1. Chapter 20: The Transfinite Universe
    4. Gödel and Computer Science
      1. Chapter 21: The Gödel Phenomenon in Mathematics: A Modern View
  14. Index