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Series:   Elements in Philosophy and Logic

Gödel's Incompleteness Theorems

Published online by Cambridge University Press:  05 April 2022

Juliette Kennedy
Juliette Kennedy
Affiliation:
University of Helsinki

Summary

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.
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Keywords

GödelFirst Incompleteness TheoremSecond Incompleteness TheoremFixed Point TheoremDiagonalization
Type
Element
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Series: Elements in Philosophy and Logic
Online ISBN: 9781108981972
Publisher: Cambridge University Press
Print publication: 14 April 2022

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