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2016 Some Remarks on Real Numbers Induced by First-Order Spectra
Sune Kristian Jakobsen, Jakob Grue Simonsen
Notre Dame J. Formal Logic 57(3): 355-368 (2016). DOI: 10.1215/00294527-3489987

Abstract

The spectrum of a first-order sentence is the set of natural numbers occurring as the cardinalities of finite models of the sentence. In a recent survey, Durand et al. introduce a new class of real numbers, the spectral reals, induced by spectra and pose two open problems associated to this class. In the present note, we answer these open problems as well as other open problems from an earlier, unpublished version of the survey.

Specifically, we prove that (i) every algebraic real is spectral, (ii) every automatic real is spectral, (iii) the subword density of a spectral real is either 0 or 1, and both may occur, and (iv) every right-computable real number between 0 and 1 occurs as the subword entropy of a spectral real.

In addition, Durand et al. note that the set of spectral reals is not closed under addition or multiplication. We extend this result by showing that the class of spectral reals is not closed under any computable operation satisfying some mild conditions.

Citation

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Sune Kristian Jakobsen. Jakob Grue Simonsen. "Some Remarks on Real Numbers Induced by First-Order Spectra." Notre Dame J. Formal Logic 57 (3) 355 - 368, 2016. https://doi.org/10.1215/00294527-3489987

Information

Received: 19 May 2011; Accepted: 9 October 2013; Published: 2016
First available in Project Euclid: 24 March 2016

zbMATH: 06621294
MathSciNet: MR3521485
Digital Object Identifier: 10.1215/00294527-3489987

Subjects:
Primary: 03C13, 68Q15, 68Q19
Secondary: 11U05, 11U09

Rights: Copyright © 2016 University of Notre Dame

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Vol.57 • No. 3 • 2016
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