Morley's theorem and Vaught's conjecture via Algebraic logic

2018/11/04 by Mohammad Assem, Assem, M., Tarek Sayed Ahmed +5
Computer Science · Mathematics · Psychology · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Philosophy and Theoretical Science

paper · pdf · doi:10.48550/arxiv.1811.02472

openalex publication_date 2018/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Vaught's Conjecture states that if T is a complete first order theory in a countable language that has more than ℵ₀ pairwise non-isomorphic countably infinite models, then T has 2^ℵ₀ such models. Morley showed that if T has more than ℵ₁ pairwise non-isomorphic countably infinite models, then it has 2^ℵ₀ such models. In this paper, we re-prove Morley's result and prove the corresponding statement for languages without equality, and for theories which are not necessarily complete. Our proof uses algebraic logic, namely the representation theory of cylindric and quasi-polyadic algebras. Also, as in Morley's proof, we use results from descriptive set theory. After all this, we show that our proof can be modified to talk about the number of models omitting a certain family of types. \endabstract

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