The isomorphism relation of theories with S-DOP in generalized Baire spaces

On Σ^1_1-complete Equivalence Relations on the Generalized Baire Space

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).

On the reducibility of isomorphism relations

We study the Borel reducibility of isomorphism relations in the generalized Baire space $κ^κ$. In the main result we show for inaccessible $κ$, that if $T$ is a classifiable theory and $T'$ is stable with OCP, then the isomorphism of models of $T$ is Borel reducible to the isomorphism of models of $T'$.

Classes of Polish spaces under effective Borel isomorphism

We study the equivalence classes under $Δ^1_1$ isomorphism, otherwise effective-Borel isomorphism, between complete separable metric spaces which admit a recursive presentation and we show the existence of strictly increasing and strictly decreasing sequences as well as of infinite antichains under the natural notion of $Δ^1_1$-reduction, as opposed to the non-effective case, where only two such classes exist, the one of the Baire space and the one of the naturals. A key tool for our study is a mapping $T \mapsto \mathcal{N}^T$ from the space of all trees on the naturals to the class of Polish…

Taking the path computably travelled

We define a real $A$ to be low for paths in Baire space (or Cantor space) if every $Π^0_1$ class with an $A$-computable element has a computable element. We prove that lowness for paths in Baire space and lowness for paths in Cantor space are equivalent and, furthermore, that these notions are also equivalent to lowness for isomorphism.

贝尔空间到紧集的分割

Partitions of Baire space into compact sets

本文在CH下构造了贝尔空间到紧集的不可摧毁分割，回答了Newelski的问题，并证明Shelah超幂模型满足a=a_T，从而在可测基数下得到ℵ₁<d<a=a_T的一致性。

On large cardinals and generalized Baire spaces

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 $E^{2,κ}_{reg}$…