Goedel's Incompleteness Theorems

The element discusses Gödel's 1931 paper, exploring the main ingredients of his proof, including arithmetization, strong representability, and the Fixed Point Theorem. It samples important proofs and examines the relationship between incompleteness and the Entscheidungsproblem. It also considers set-theoretical incompleteness and philosophical consequences.

Format: Paperback / softback
Length: 75 pages
Publication date: 14 April 2022
Publisher: Cambridge University Press

This Element takes a deep dive into Gödel's 1931 paper, presenting the first presentation of the Incompleteness Theorems. It explores passages that may puzzle students, such as the enigmatic footnote 48a. It examines the main ingredients of Gödel's proof, including arithmetization, strong representability, and the Fixed Point Theorem, in a layered fashion. It revisits these aspects: semantic, syntactic, computational, philosophical, and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, such as those due to Kuratowski, Smullyan, and Robinson, as well as newer proofs of other independent statements due to H. Friedman, Weiermann, and Paris-Harrington. It examines the question of whether the incompleteness of Peano Arithmetic immediately gives rise to the undecidability of the Entscheidungsproblem, as recently argued by Kripke. It also considers set-theoretical incompleteness and explores some of the philosophical consequences considered in the literature.

ISBN-13: 9781108986991
Edition number: New ed