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A descriptive Main Gap Theorem

Francesco Mangraviti

Institut für Philosophie I, Ruhr Universität Bochum, Universitätsstr. 150, 44801 Bochum, Germany

E-mail Address: Francesco.Mangraviti@ruhr-uni-bochum.de

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Luca Motto Ros

Dipartimento di matematica «Giuseppe Peano», Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

E-mail Address: luca.mottoros@unito.it

Corresponding author.

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https://doi.org/10.1142/S0219061320500257Cited by:2 (Source: Crossref)

Abstract

Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc.230(1081) (2014) 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory $T$ $T$ and the Borel rank of the isomorphism relation $≅_{T}^{κ}$ $≅_{T}^{κ}$ on its models of size $κ$ $κ$ , for $κ$ $κ$ any cardinal satisfying $κ^{< κ} = κ > 2_{0}^{ℵ_{0}}$ $κ^{< κ} = κ > 2^{ℵ_{0}}$ . This is achieved by establishing a link between said rank and the $ℒ_{\infty κ}$ -Scott height of the $κ$ -sized models of $T$ , and yields to the following descriptive set-theoretical analog of Shelah’s Main Gap Theorem: Given a countable complete first-order theory $T$ , either $≅_{T}^{κ}$ is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is $κ^{+} > ℵ_{1}$ ), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah’s theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of $≅_{T}^{κ}$ , and provide a characterization of categoricity of $T$ in terms of the descriptive set-theoretical complexity of $≅_{T}^{κ}$ .

Keywords:

AMSC: 03E15, 03C45

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