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First and second Gödel’s theorems and related results

  • Duccio Pianigiani

We start with the central result from which we derive the first Gödel Theorem. That the 1931 proof of the first theorem is syntactic and constructive. This means that it doesn’t appeal to truth and that we concretely give examples of sentences that are independent (neither provable nor refutable). We show Rosser’s version of the first incompleteness theorem. Then we discuss the limit of incompleteness phenomenon and we illustrate a complete and decidable theory. We introduce nonostandard model and we take another look at incompleteness: Tennenbaum’s theorem.

  • Keywords:
  • first and second incompleteness theorems,
  • Presburger arithmetic,
  • Rosser sentences,
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Duccio Pianigiani

University of Siena, Italy - ORCID: 0000-0001-9441-7226

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  • Publication Year: 2025
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Chapter Information

Chapter Title

First and second Gödel’s theorems and related results

Authors

Duccio Pianigiani

Language

English

DOI

10.36253/979-12-215-0778-2.08

Peer Reviewed

Publication Year

2025

Copyright Information

© 2025 Author(s)

Content License

CC BY-SA 4.0

Metadata License

CC0 1.0

Bibliographic Information

Book Title

Lectures in Proof Theory and Complexity

Authors

Duccio Pianigiani

Peer Reviewed

Publication Year

2025

Copyright Information

© 2025 Author(s)

Content License

CC BY-SA 4.0

Metadata License

CC0 1.0

Publisher Name

Firenze University Press, USiena Press

DOI

10.36253/979-12-215-0778-2

eISBN (pdf)

979-12-215-0778-2

eISBN (xml)

979-12-215-0779-9

Series Title

UNIverSI. Ricerca e Didattica all’Università di Siena

Series ISSN

3035-5915

Series E-ISSN

3035-5931

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