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  3. Gödel's Incompleteness Theorems
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    • Publisher:
      Cambridge University Press
      Publication date:
      05 April 2022
      14 April 2022
      ISBN:
      9781108981972
      9781108986991
      DOI:
      https://doi.org/10.1017/9781108981972
      Dimensions:
      Weight & Pages:
      Dimensions:
      (229 x 152 mm)
      Weight & Pages:
      0.15kg, 90 Pages
      • Series:
      Elements in Philosophy and Logic
      Subjects:
      Logic, Philosophy
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    Series:
    Elements in Philosophy and Logic
    Subjects:
    Logic, Philosophy
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    Book description

    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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