(EBook PDF) Godels Theorem A Very Short Introduction 1st edition by Moore 0192663585 9780192663580 full chapters

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ISBN-10 ‏ : ‎ 0192663585 

ISBN-13 ‏ : ‎9780192663580

Author: A. W. Moore

Kurt Gödel first published his celebrated theorem, showing that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic, nearly a century ago. The theorem challenged prevalent presuppositions about the nature of mathematics and was consequently of considerable mathematical interest, while also raising various deep philosophical questions. Gödel’s Theorem has since established itself as a landmark intellectual achievement, having a profound impact on today’s mathematical ideas. Gödel and his theorem have attracted something of a cult following, though his theorem is often misunderstood.

Gödel’s Theorem: A Very Short Introduction 1st Table of contents:

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)

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