This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
By:
Juliette Kennedy (University of Helsinki) Imprint: Cambridge University Press Country of Publication: United Kingdom Edition: New edition Dimensions:
Height: 229mm,
Width: 151mm,
Spine: 5mm
Weight: 150g ISBN:9781108986991 ISBN 10: 1108986994 Series:Elements in Philosophy and Logic Pages: 75 Publication Date:14 April 2022 Audience:
General/trade
,
ELT Advanced
Format:Paperback Publisher's Status: Active
Introduction; 1. The first version of the proof; 2. Gödel's 'intuitionistically acceptable' second proof of the First Incompleteness Theorem; 3. The unprovability of consistency; 4. Löb conditions and adequacy; 5. Other proofs of the First and Second Theorems; 6. Mathematical Incompleteness; 7. Set Theoretical Incompleteness; 8. Further philosophical consequences of the Incompleteness Theorems.
