arXiv subjects
Publications and source records attributed to Vadim Kulikov.
It is shown that the power set of $κ$ ordered by the subset relation modulo various versions of the non-stationary deal can be embedded into the partial order of Borel equivalence relations on $2^κ$ under Borel reducibility. Here $κ$ is uncountable regular cardinal with $κ^{<κ} = κ$.
Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.
Using methods of descriptive theory it is shown that the classification problem for wild knots is strictly harder than that for countable structures.
The paper deals with the program of determining the complexity of various homeomorphism relations. The homeomorphism relation on compact Polish spaces is known to be reducible to an orbit equivalence relation of a continuous Polish group action (Kechris-Solecki). It is shown that this result extends to locally compact Polish spaces, but does not hold for spaces in which local compactness fails at only one point. In fact it fails for those subsets of $\mathbb{R}^3$ which are unions of an open set and a point. In the end a list of open problems is given in this area of research.
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 ($κ>ω$). 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 $V\ne L$ and prove $Σ_1^1$-completeness results for weakly ineffable and weakly compact $κ$.
We study probabilistic team semantics which is a semantical framework allowing the study of logical and probabilistic dependencies simultaneously. We examine and classify the expressive power of logical formalisms arising by different probabilistic atoms such as conditional independence and different variants of marginal distribution equivalences. We also relate the framework to the first-order theory of the reals and apply our methods to the open question on the complexity of the implication problem of conditional independence.
Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal $κ$. We show the consistency of $E^{λ^{++},λ^{++}}_{λ\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_λ$ in the space $(λ^{++})^{λ^{++}}$, being continuously reducible to $E^{2,λ^{++}}_{λ^+\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_{λ^+}$ in the space $2^{λ^{++}}$. Then we show the consistency of $E^{2,κ}_{reg}$, the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space $2^κ$, being $Σ_1^1$-complete. We finish by showing, for $Π_2^1$-indescribable $κ$, that the isomorphism relation between dense linear orders of cardinality $κ$ is $Σ_1^1$-complete.
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 are considered on models of some fixed uncountable cardinality obeying certain restrictions.
We investigate regularity properties derived from tree-like forcing notions in the setting of "generalized descriptive set theory", i.e., descriptive set theory on $κ^κ$ and $2^κ$, for regular uncountable cardinals $κ$.
We prove results that falsify Silver's dichotomy for Borel equivalence relations on the generalised Baire space under the assumption V=L.
In this paper we study the Borel reducibility of Borel equivalence relations, including some orbit equivalence relations, on the generalised Baire space $κ^κ$ for an uncountable $κ$ with the property $κ^{<κ}=κ$. The theory looks quite different from its classical counterpart where $κ=ω$, although some basic theorems do generalise.
The space $F(\ell_2)$ of all closed subsets of $\ell_2$ is a Polish space. We show that the subset $P\subset F(\ell_2)$ consisting of the purely 1-unrectifiable sets is $\Pii$-complete.
Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if V = L, then many of them are Σ^1_1-complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in ZFC whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is Σ^1_1-complete (it is, if V = L, but can be forced not to be).
We start by giving a survey to the theory of Borel*(κ) sets in the generalized Baire space Baire(κ) = κ^κ. In particular we look at the relation of this complexity class to other complexity classes which we denote by Borel(κ), Δ^1_1(κ) and Σ^1_1(κ) and the connections between Borel*(κ)-sets and the infinitely deep language M_{κ^+κ}. In the end of the paper we prove the consistency of Borel*(κ) \ne Σ^1_1(κ).