Non-isomorphism invariant Borel quantifiers

Abstract:

Every isomorphism invariant Borel subset of the space of structures on the natural numbers in a countable relational language is definable in ${\mathscr {L}}_{\omega _1\omega }$ by a theorem of Lopez-Escobar. We derive variants of this result for stabilizer subgroups of the symmetric group ${\mathrm {Sym}}(\mathbb {N})$ for families of relations and non-isomorphism invariant generalized quantifiers on the natural numbers such as “for all even numbers”. Moreover we produce a binary quantifier $Q$ for every closed subgroup of ${\mathrm {Sym}}(\mathbb {N})$ such that the Borel sets of structures invariant under the subgroup action are exactly the sets of structures definable in ${\mathscr {L}}_{\omega _1\omega }(Q)$.