Gödel's Incompleteness Theorems

Erscheinungsjahr 2022
ISBN: 978-1-108-98699-1
Verlag: Cambridge University Press

Buch, Englisch, 75 Seiten, Format (B × H): 152 mm x 229 mm, Gewicht: 145 g

Reihe: Elements in Philosophy and Logic

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.

Autoren/Hrsg.

Kennedy, Juliette

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik
• Geisteswissenschaften Philosophie Philosophische Logik, Argumentationstheorie

Inhaltsverzeichnis

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.