Socrates did it before Gödel

Josef Wolfgang Degen

DOI: http://dx.doi.org/10.12775/LLP.2011.011

Abstract


We translate Socrates’ famous saying I know that I know nothing into the arithmetical sentence I prove that I prove nothing. Then it is easy to show that this translated saying is formally undecidable in formal arithmetic, using Gödel’s Second Incompleteness Theorem. We investigate some variations of this Socrates-Gödel sentence. In an appendix we sketch a ramified epistemic logic with propositional quantifiers in order to analyze the Socrates-Gödel sentence in a more logical way, separated from the arithmetical context.

Keywords


the “paradoxon” of Socrates; Gödel’s Second Incompleteness Theorem; a ramified epistemic logic with propositional quantifiers

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References


Kurt Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik 38 (1931): 173–198.

Craig Smorynski, “The incompleteness theorems”, in: Jon Barwise (editor), Handbook of Mathematical Logic, North-Holland, 1977.

Craig Smorynski, Self-Reference and Modal Logic, Springer, 1985.








Print ISSN: 1425-3305
Online ISSN: 2300-9802

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