Mathematical Logic Quarterly 58 (4-5):307-316 (2012)
| Abstract |
By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We shall show also that Vopěnka's proof can be reformulated in arithmetic by using the arithmetized completeness theorem
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| Keywords | The Incompleteness Theorems Berry's Paradox Kolmogorov Complexity |
| Categories | (categorize this paper) |
| DOI | 10.1002/malq.201110067 |
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References found in this work BETA
Mathematical Logic as Based on the Theory of Types.Bertrand Russell - 1908 - American Journal of Mathematics 30 (3):222-262.
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.K. Gödel - 1931 - Monatshefte für Mathematik 38 (1):173--198.
A Note on Boolos' Proof of the Incompleteness Theorem.Makoto Kikuchi - 1994 - Mathematical Logic Quarterly 40 (4):528-532.
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Citations of this work BETA
Incompleteness Via Paradox and Completeness.Walter Dean - 2020 - Review of Symbolic Logic 13 (3):541-592.
On the Diagonal Lemma of Gödel and Carnap.Saeed Salehi - 2020 - Bulletin of Symbolic Logic 26 (1):80-88.
On Constructivity and the Rosser Property: A Closer Look at Some Gödelean Proofs.Saeed Salehi & Payam Seraji - 2018 - Annals of Pure and Applied Logic 169 (10):971-980.
Liar-Type Paradoxes and the Incompleteness Phenomena.Makoto Kikuchi & Taishi Kurahashi - 2016 - Journal of Philosophical Logic 45 (4):381-398.
Rosser-Type Undecidable Sentences Based on Yablo’s Paradox.Taishi Kurahashi - 2014 - Journal of Philosophical Logic 43 (5):999-1017.
View all 6 citations / Add more citations
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