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Logic and Logical Philosophy

A Note on Gödel’s First Disjunct Formalised in DTK System
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A Note on Gödel’s First Disjunct Formalised in DTK System

Authors

  • Antonella Corradini Department of Psychology, Università Cattolica del Sacro Cuore Milano https://orcid.org/0000-0001-8227-2226
  • Sergio Galvan Department of Philosophy, Università Cattolica del Sacro Cuore Milano https://orcid.org/0000-0001-8552-1099

DOI:

https://doi.org/10.12775/LLP.2024.025

Keywords

Penrose’s second argument, Gödel’s disjunction, DT system, DTK system, computational model of mind, arguments in favour of the first horn of Gödel’s disjunction

Abstract

This note clarifies the significance of the proof of Gödel’s first disjunct obtained through the formalisation of Penrose’s second argument within the DTK system. It analyses two formulations of the first disjunct – one general and the other restricted – and dwells on the demonstration of the restricted version, showing that it yields the following result: if by F we denote the set of propositions derivable from any formalism and by K the set of mathematical propositions humanly knowable, then, given certain conditions, F necessarily differs from K. Thus it is possible that K surpasses F but also, on the contrary, that F surpasses K. In the latter case, however, the consistency of F is humanly undecidable.

References

Corradini A., and S. Galvan, 2022, “Analysis of Penrose’s second argument formalised in DTK system”, Logic and Logical Philosophy, 31: 471-500. DOI: http://dx.doi.org/10.12775/LLP.2021.019

Feferman, S., 2008, “Axioms for determinatess and truth”, The Journal of Symbolic Logic, 1(2), 204–217. DOI: http://dx.doi.org/10.1017/S1755020308080209

Gödel, K., 1995, “Some basic theorems on the foundations of mathematics and theirimplications” (1951), pages 304–323 in S. Feferman et al. (eds.), Collected Works, Volume III: Unpublished Essays and Lectures, New York: Oxford University Press.

Koellner, P., 2016, “Gödel’s disjunction”, pages 148–188 in L. Horsten and P. Welch (eds.), Gödel’s Disjunction: The Scope and Limits of Mathematical Knowledge. New York: Oxford University Press. DOI: http://dx.doi.org/10.1093/acprof:oso/9780198759591.003.0007

Koellner, P., 2018a, “On the question of whether the mind can be mechanized, I: From Gödel to Penrose”, The Journal of Philosophy, CXV(7): 337–360. DOI: http://dx.doi.org/10.5840/jphil2018115721

Koellner, P., 2018b, “On the question of whether the mind can be mechanized, II: Penrose’s New Argument”, The Journal of Philosophy, CXV(7): 453–484. DOI: http://dx.doi.org/10.5840/jphil2018115926

Penrose, R., 1994, Shadows of the Mind: A Search for the Missing Science of Consciousness, Oxford University Press. DOI: http://dx.doi.org/10.1093/oso/9780195106466.001.0001

Penrose, R., 1996, “Beyond the doubting of a shadow. A reply to commentaries on Shadows of the Mind”. https://calculemus.org/MathUniversalis/NS/10/01penrose.html

Shapiro, S., 2016, “Idealization, mechanism, and knowability”, pages 189–207 in L. Horsten and P. Welch (eds.), Gödel’s Disjunction: The Scope and Limits of Mathematical Knowledge, New York: Oxford University Press. DOI: http://dx.doi.org/10.1093/acprof:oso/9780198759591.003.0008

Logic and Logical Philosophy

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Published

2024-10-23

How to Cite

1.
CORRADINI, Antonella and GALVAN, Sergio. A Note on Gödel’s First Disjunct Formalised in DTK System. Logic and Logical Philosophy. Online. 23 October 2024. Vol. 33, no. 4, pp. 555-565. [Accessed 5 July 2025]. DOI 10.12775/LLP.2024.025.
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Issue

Vol. 33 No. 4 (2024): December

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Articles

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Copyright (c) 2024 Antonella Corradini, Sergio Galvan

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This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

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