The complexity of the embeddability relation between torsion-free abelian groups of uncountable size

We prove that for every uncountable cardinal $\kappa$ such that $\kappa^{<\kappa}=\kappa$, the quasi-order of embeddability on the $\kappa$-space of $\kappa$-sized graphs Borel reduces to the embeddability on the $\kappa$-space of $\kappa$-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability on the $\kappa$-space of $\kappa$-sized $R$-modules, for every $\mathbb{S}$-cotorsion-free ring $R$ of cardinality less than the continuum. As a consequence we get that all the previous are complete $\boldsymbol{\Sigma}^1_1$ quasi-orders.Comment: 14 pages, final versio

A generalized Borel-reducibility counterpart of Shelah's Main Gap Theorem

Peer reviewe

Finding the Main Gap in the Borel-reducibility Hierarchy

Model theory and set theory are two disciplines of mathematical logic which can be used to study the Borel reducibility hierarchy in the generalized Baire space. These two disciplines are connected when the complexity of complete first order theories is studied. Each of these disciplines has its approach to measure the complexity of complete first order theories. The Borel reducibility hierarchy in the generalized Baire space shows us a deep connection between these two approaches. In Shelah's stability theory, a classifiable theory is a theory with an invariant that determines the structures up to isomorphisms, a theory with no invariant of this kind is a non-classifiable theory. This tell us that a theory with an invariant of this kind is less complex than a theory with no invariant of this kind. Shelah's stability theory tells us that every countable complete first-order classifiable theory is less complex than all countable complete first-order non-classifiable theories. The subject of study in this thesis is the question: Are all classifiable theories less complex than all the non-classifiable theories, in the Borel reducibility hierarchy? There are two frames where this question can be studied, the generalized Baire space and the generalized Cantor space. It is known that for every theory T, the isomorphism relation of T in the generalized Cantor space and the isomorphism relation of T in the generalized Baire space have the same complexity. This gives us the freedom to choose in which space we would like to work. This question was studied by Friedman, Hyttinen, and Kulikov between others, in previous works. Some of the results in those works pointed out that the relation equivalence modulo the non-stationary ideal might be one of the keys to understand the reducibility of the isomorphism relations. The work of Friedman, Hyttinen, and Kulikov leads to two approaches for the main question: Is it provable in ZFC that in the generalized Cantor space, the isomorphism relation of T is Borel reducible to the equivalence modulo the non-stationary ideal, for T a classifiable theory? Is it provable in ZFC that in the generalized Cantor space, the equivalence modulo the non-stationary ideal is Borel reducible to the isomorphism relation of T, for T a non-classifiable theory? Is it provable in ZFC that in the generalized Baire space, the isomorphism relation of T is Borel reducible to the equivalence modulo the non-stationary ideal, for T a classifiable theory? Is it provable in ZFC that in the generalized Baire space, the equivalence modulo the non-stationary ideal is Borel reducible to the isomorphism relation of T, for T a non-classifiable theory? The work of Friedman, Hyttinen, and Kulikov gives a partial answer to this question. At the same time this points out to a question that might be the key to understand the the connection between classification theory and the Borel reducibility hierarchy: Does the equivalence modulo the non-stationary ideal has the same complexity in the generalized Cantor space as in the generalized Baire space? It is known that the isomorphism relations have the same complexity in the generalized Cantor space as in the generalized Baire space. These are the questions studied in this thesis

Shelah's Main Gap and the generalized Borel-reducibility

We answer one of the main questions in generalized descriptive set theory, Friedman-Hyttinen-Kulikov conjecture on the Borel-reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel-reducibility notions of complexity. For any $\kappa$ satisfying $\kappa=\lambda^+=2^\lambda$ and $2^{\mathfrak{c}}\leq\lambda=\lambda^{\omega_1}$, we show that if $T$ is a classifiable theory and $T'$ not, then the isomorphism of models of $T'$ is strictly above the isomorphism of models of $T$ with respect to Borel-reducibility. We also show that the following can be forced: for any countable first-order theory in a countable vocabulary, $T$, the isomorphism of models of $T$ is either $\Delta^1_1$ or analytically-complete