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The number of countable models via Algebraic logic

Assem, Mohammad and Sayed-Ahmed, Tarek and Sági, Gábor and Sziráki, Dorottya (2013) The number of countable models via Algebraic logic. Other. Kézirat. (Unpublished)

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Abstract

Vaught's Conjecture states that if T is a complete First order theory in a countable language that has more than aleph_0 pairwise non isomorphic countable models, then T has 2^aleph_0 such models. Morley showed that if T has more than aleph_1 pairwise non isomorphic countable models, then it has 2^aleph_0 such models. In this paper, we First show how we can use algebraic logic, namely the representation theory of cylindric and quasi-polyadic algebras, to study Vaught's conjecture (count models), and we re-prove Morley's above mentioned theorem. Second, we show that Morley's theorem holds for the number of non isomorphic countable models omitting a countable family of types. We go further by giving examples showing that although this number can only take the values given by Morley's theorem, it can be different from the number of all non isomorphic countable models. Moreover, our examples show that the number of countable models omitting a family of types can also be either aleph_1 or 2 and therefore different from the possible values provided by Vaught's conjecture and by his well known theorem; in the case of aleph_1, however, the family is uncountable. Finally, we discuss an omitting types theorem of Shelah.

Item Type: Monograph (Other)
Subjects: Q Science / természettudomány > QA Mathematics / matematika
Depositing User: Dr Gábor Sági
Date Deposited: 26 Sep 2014 07:30
Last Modified: 26 Sep 2014 07:30
URI: http://real.mtak.hu/id/eprint/16985

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