Publications with Vadim Kulikov

3. Tapani Hyttinen, Vadim Kulikov, Miguel Moreno, On Σ_1^1completeness of Quasiorders on K^K. Fundamenta Mathematicae (2020) 251: 245  268.

Abstract: We prove under V=L that the inclusion modulo the nonstationary ideal is a Σ^1_1complete quasiorder in the generalized Borelreducibility hierarchy (K > ω). This improvement to known results in L has many new consequences concerning the Σ^1_1completeness of quasiorders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the nonstationary 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 firstorder theory (not necessarily complete) is not Δ^1_1, then it is Σ^1_1complete. We also study the case V is different from L and prove Σ^1_1completeness 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 Borelreducibility between equivalence relations modulo restrictions of the nonstationary ideal on some fixed cardinal K. We show the consistency of E^(λ++,λ++)_(λclub), the relation of equivalence modulo the nonstationary 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 nonstationary 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 nonstationary ideal restricted to regular cardinals in the space 2^K, is Σ^1_1complete. We finish by showing, for Π^1_2indescribable K, that the isomorphism relation between dense linear orders of cardinality K is Σ^1_1complete.


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1. Tapani Hyttinen, Vadim Kulikov, Miguel Moreno, A generalized Borelreducibility counterpart of Shelah's main gap theorem. Archive for Mathematical Logic (2017) 56: 175  185.

Abstract: We study the Borelreducibility 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 Borelreducibility. In fact, we can also ensure that a range of equivalence relations modulo various nonstationary ideals are strictly between those isomorphism relations. The isomorphism relations areconsidered on models of some fixed uncountable cardinality obeying certain restrictions.


PDF  arXiv  Journal