Miguel Moreno

Postdoctoral researcher, Institute of Mathematics, University of Vienna.

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Publications with Vadim Kulikov

3. Tapani Hyttinen, Vadim Kulikov, Miguel Moreno, On Σ_1^1-completeness of Quasi-orders on K^K. Fundamenta Mathematicae (2020) 251: 245 -- 268.
Abstract: We prove under V=L that the inclusion modulo the non-stationary ideal is a Σ^1_1-complete quasi-order in the generalized Borel-reducibility hierarchy (K > ω). This improvement to known results in L has many new consequences concerning the Σ^1_1-completeness of quasi-orders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the non-stationary ideal. This serves as a partial or complete answer to several open problems stated in literature. Additionally the theorem is applied to prove a dichotomy in L: If the isomorphism of a countable first-order theory (not necessarily complete) is not Δ^1_1, then it is Σ^1_1-complete. We also study the case V is different from L and prove Σ^1_1-completeness results for weakly ineffable and weakly compact K.
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2. David Asperó, Tapani Hyttinen,Vadim Kulikov, Miguel Moreno, Reducibility of equivalence relations arising from nonstationary ideals under large cardinal assumptions. Notre Dame Journal of Formal Logic (2019) 60: 665 -- 682.
Abstract: Working under large cardinal assumptions such as supercompactness, westudy the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal K. We show the consis-tency of E^(λ++,λ++)_(λ-club), the relation of equivalence modulo the non-stationary ideal restricted to S^(λ++)_λ in the space (λ++)^(λ++), being continuously reducible to E^(2,λ;++)_(λ+-club), the relation of equivalence modulo the non-stationary ideal restricted to S^(λ++)_(λ+) in the space 2^(λ++). Then we show that for K ineffable E^(2,K)_(reg), the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space 2^K, is Σ^1_1-complete. We finish by showing, for Π^1_2-indescribable K, that the isomorphism relation between dense linear orders of cardinality K is Σ^1_1-complete.
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1. Tapani Hyttinen, Vadim Kulikov, Miguel Moreno, A generalized Borel-reducibility counterpart of Shelah's main gap theorem. Archive for Mathematical Logic (2017) 56: 175 -- 185.
Abstract: We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations areconsidered on models of some fixed uncountable cardinality obeying certain restrictions.
PDF - arXiv - Journal