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 L ω 1 ω {\mathscr {L}}_{\omega _1\omega } by a theorem of Lopez-Escobar. We derive variants of this result for stabilizer subgroups of the symmetric group S y m ( N ) {\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 Q for every closed subgroup of S y m ( N ) {\mathrm {Sym}}(\mathbb {N}) such that the Borel sets of structures invariant under the subgroup action are exactly the sets of structures definable in L ω 1 ω ( Q ) {\mathscr {L}}_{\omega _1\omega }(Q) .