Gödel’s Theorem.. A Very Short Introduction

✍ Scribed by A. W. Moore

• Tongue: English
• Leaves: 137
• Category: Library

✦ Table of Contents

Halftitle page
Series page
Title page
Copyright page
Dedication page
Contents
Preface
Note on the Appendix
List of illustrations
Chapter 1. What is Gödel’s theorem?
Arithmetic
Axiomatization
Proof
Arithmetical truths and falsehoods
Misconceptions of Gödel’s theorem
Gödel’s two theorems
Chapter 2. Axiomatizationits appeal and demands
The general appeal of axiomatization
The mathematical appeal of axiomatization
Euclid’s Elements: a (flawed) paradigm
Independence and consistency
Justifying the assumption of consistency
Chapter 3. Historical background
Frege’s project
The collapse of Frege’s project
Principia Mathematica
Hilbert’s programme
Chapter 4. The key concepts involved in Gödel’s theorem
Formal languages
Logical and non-logical vocabulary
Key concepts
A new statement of Gödel’s theorem
Gödel numbering
Chapter 5. The diagonal proof of Gödel’s theorem
The semantic version of the proof
Three noteworthy features of the proof
The non-semantic version of the proof
Chapter 6. A second proof of Gödel’s theorem, and a proof of Gödel’s second theorem
The axiomatizability of PA
The sufficient strength of PA
The soundness of PA
A sketch of Gödel’s proof
A closer look at the construction of Π†i(i)
The second theorem
Chapter 7. Hilbert’s programme, the human mind, and computers
Threats to Hilbert’s programme
Self-consciousness
The Lucas–Penrose argument
Responses to the Lucas–Penrose argument
Chapter 8. Making sense in and of mathematics
Acknowledging consistency
A challenge presented by Gödel’s theorem
Meaning and understanding (in mathematics)
Appendix
Proof
Proof
References
Further reading
Index
Numbers
German Philosophy