On large cardinals and generalized Baire spaces

TL;DR

This paper explores the complexity of equivalence relations and isomorphism relations in generalized Baire spaces under large cardinal assumptions, establishing consistency results and completeness classifications.

Contribution

It demonstrates the consistency of certain reducibility and completeness results for equivalence relations in large cardinal contexts, advancing the understanding of their descriptive set-theoretic complexity.

Findings

Equivalence relations modulo non-stationary ideals can be continuously reduced under large cardinal assumptions.

Certain equivalence relations are shown to be $oldsymbol{ ext{Σ}}_1^1$-complete in generalized Baire spaces.

The isomorphism relation for dense linear orders of size $oldsymbol{ ext{kappa}}$ is $oldsymbol{ ext{Σ}}_1^1$-complete for $oldsymbol{ ext{Π}}_2^1$-indescribable $oldsymbol{ ext{kappa}}$.

Abstract

Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal $\kappa$. We show the consistency of $E^{\lambda^{++},\lambda^{++}}_{\lambda\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{\lambda^{++}}_\lambda$ in the space $(\lambda^{++})^{\lambda^{++}}$, being continuously reducible to $E^{2,\lambda^{++}}_{\lambda^+\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{\lambda^{++}}_{\lambda^+}$ in the space $2^{\lambda^{++}}$. Then we show the consistency of $E^{2,\kappa}_{reg}$, the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space $2^{\kappa}$, being $\Sigma_1^1$-complete. We finish by showing, for $\Pi_2^1$-indescribable $\kappa$, that the isomorphism relation between dense linear orders of cardinality $\kappa$ is $\Sigma_1^1$-complete.