On $\Sigma_1^1$-completeness of quasi-orders on $\kappa^\kappa$

TL;DR

Under the assumption V=L, the paper establishes that the inclusion modulo the non-stationary ideal is a $\, ext{Sigma}_1^1$-complete quasi-order in the generalized Borel hierarchy, impacting the understanding of complexity of various mathematical structures.

Contribution

It improves known results by proving $\, ext{Sigma}_1^1$-completeness of certain quasi-orders under V=L and explores consequences for isomorphism relations and other equivalence relations.

Findings

Inclusion modulo non-stationary ideal is $\, ext{Sigma}_1^1$-complete under V=L.

A dichotomy in L: non-$\, ext{Delta}_1^1$ isomorphisms are $\, ext{Sigma}_1^1$-complete.

$\, ext{Sigma}_1^1$-completeness results for weakly ineffable and weakly compact $\, ext{kappa}$.

Abstract

We prove under $V=L$ that the inclusion modulo the non-stationary ideal is a $\Sigma_1^1$-complete quasi-order in the generalized Borel-reducibility hierarchy ($\kappa>\omega$). This improvement to known results in $L$ has many new consequences concerning the $\Sigma_1^1$-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 $\Delta_1^1$, then it is $\Sigma_1^1$-complete. We also study the case $V\ne L$ and prove $\Sigma_1^1$-completeness results for weakly ineffable and weakly compact $\kappa$.