Non-permutation invariant Borel quantifiers

TL;DR

This paper extends Lopez-Escobar's theorem to fixed relations and non-permutation invariant quantifiers, characterizing G-invariant sets via definability with specific quantifiers.

Contribution

It introduces variants of the Lopez-Escobar theorem for non-permutation invariant quantifiers and constructs quantifiers characterizing G-invariant sets.

Findings

Characterization of G-invariant sets using specific quantifiers

Extension of Lopez-Escobar theorem to non-permutation invariants

Existence of a closed binary quantifier for each closed subgroup G

Abstract

Every permutation invariant Borel subset of the space of countable structures is definable in $\La_{\omega_1\omega}$ by a theorem of Lopez-Escobar. We prove variants of this theorem relative to fixed relations and fixed non-permutation invariant quantifiers. Moreover we show that for every closed subgroup $G$ of the symmetric group $S_{\infty}$, there is a closed binary quantifier $Q$ such that the $G$-invariant subsets of the space of countable structures are exactly the $\La_{\omega_1\omega}(Q)$-definable sets.