Fulfillability, Instability, and Incompleteness
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Roy T Cook
Abstract
The purpose of this essay is to publicize, and to a more limited extent, further develop, an alternate proof of Gödel’s incompleteness theorem due to Saul Kripke, based on a notion called fulfillability. Kripke’s work has been publicized in talks, but at the time of writing this essay the only published discussion of the material appears in Putnam (2000). Here, a more detailed and more accessible overview of the approach is given, centered on a novel generalization - the Instability Theorem. After setting up the technical machinery and demonstrating the Instability Theorem, we prove both the first incompleteness theorem and Lob’s theorem. We conclude with some observations regarding potential directions for future research along these lines.
Abstract
The purpose of this essay is to publicize, and to a more limited extent, further develop, an alternate proof of Gödel’s incompleteness theorem due to Saul Kripke, based on a notion called fulfillability. Kripke’s work has been publicized in talks, but at the time of writing this essay the only published discussion of the material appears in Putnam (2000). Here, a more detailed and more accessible overview of the approach is given, centered on a novel generalization - the Instability Theorem. After setting up the technical machinery and demonstrating the Instability Theorem, we prove both the first incompleteness theorem and Lob’s theorem. We conclude with some observations regarding potential directions for future research along these lines.
Chapters in this book
- Frontmatter I
- Contents V
- List of Abbreviations VII
- An Introduction to Hilary Putnam 1
- Introduction to this Volume 47
- Putnam’s Proof Revisited 63
- Language, Meaning, and Context Sensitivity: Confronting a “Moving-Target” 89
- Externalism and the First-Person Perspective 107
- Putnam on Trans-Theoretical Terms and Contextual Apriority 131
- Mathematical Internal Realism 157
- The Labyrinth of Quantum Logic 183
- Fulfillability, Instability, and Incompleteness 207
- Putnam’s Aristotle 227
- Davidson and Putnam on the Antinomy of Free Will 249
- Putnam on Radical Scepticism: Wittgenstein, Cavell, and Occasion- Sensitive Semantics 263
- Natural Laws and Human Language 289
- Balance in The Golden Bowl: Attuning Philosophy and Literary Criticism 309
- Bibliography 331
- Contributors 349
- Index 353
Chapters in this book
- Frontmatter I
- Contents V
- List of Abbreviations VII
- An Introduction to Hilary Putnam 1
- Introduction to this Volume 47
- Putnam’s Proof Revisited 63
- Language, Meaning, and Context Sensitivity: Confronting a “Moving-Target” 89
- Externalism and the First-Person Perspective 107
- Putnam on Trans-Theoretical Terms and Contextual Apriority 131
- Mathematical Internal Realism 157
- The Labyrinth of Quantum Logic 183
- Fulfillability, Instability, and Incompleteness 207
- Putnam’s Aristotle 227
- Davidson and Putnam on the Antinomy of Free Will 249
- Putnam on Radical Scepticism: Wittgenstein, Cavell, and Occasion- Sensitive Semantics 263
- Natural Laws and Human Language 289
- Balance in The Golden Bowl: Attuning Philosophy and Literary Criticism 309
- Bibliography 331
- Contributors 349
- Index 353
