The 𝜔-Vaught’s conjecture

Gonzalez, David; Montalbán, Antonio

doi:10.1090/tran/8950

The $\omega$-Vaught’s conjecture
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by David Gonzalez and Antonio Montalbán;

Trans. Amer. Math. Soc. 376 (2023), 5989-6008

DOI: https://doi.org/10.1090/tran/8950

Published electronically: May 31, 2023

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Abstract:

We introduce the $\omega$-Vaught’s conjecture, a strengthening of the infinitary Vaught’s conjecture. We believe that if one were to prove the infinitary Vaught’s conjecture in a structural way without using techniques from higher recursion theory, then the proof would probably be a proof of the $\omega$-Vaught’s conjecture. We show the existence of an equivalent condition to the $\omega$-Vaught’s conjecture and use this tool to show that all infinitary sentences whose models are linear orders satisfy the $\omega$-Vaught’s conjecture.

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