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January 2021 Revisiting Chaitin’s Incompleteness Theorem
Christopher P. Porter
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Notre Dame J. Formal Logic 62(1): 147-171 (January 2021). DOI: 10.1215/00294527-2021-0006

Abstract

In the mid-1970s, Gregory Chaitin proved a novel incompleteness theorem, formulated in terms of Kolmogorov complexity, a measure of complexity that features prominently in algorithmic information theory. Chaitin further claimed that his theorem provides insight into both the source and scope of incompleteness, a claim that has been subject to much criticism. In this article, I consider a new strategy for vindicating Chaitin’s claims, one informed by recent work of Bienvenu, Romashchenko, Shen, Taveneaux, and Vermeeren that extends and refines Chaitin’s incompleteness theorem. As I argue, this strategy, though more promising than previous attempts, fails to vindicate Chaitin’s claims. Lastly, I will suggest an alternative interpretation of Chaitin’s theorem, according to which the theorem indicates a trade-off that comes from working with a sufficiently strong definition of randomness—namely, that we lose the ability to certify randomness.

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Christopher P. Porter. "Revisiting Chaitin’s Incompleteness Theorem." Notre Dame J. Formal Logic 62 (1) 147 - 171, January 2021. https://doi.org/10.1215/00294527-2021-0006

Information

Received: 19 September 2019; Accepted: 6 May 2020; Published: January 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1215/00294527-2021-0006

Subjects:
Primary: 03F40
Secondary: 03A05, 03D32

Rights: Copyright © 2021 University of Notre Dame

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Vol.62 • No. 1 • January 2021
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