Non-isomorphism Invariant Borel Quantifiers

Every isomorphism invariant Borel subset of the space of structures on the natural numbers in a countable relational language is definable in Lω1ω by a theorem of Lopez-Escobar. We derive variants of this result for stabilizer subgroups of the symmetric group Sym(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 Sym(N) such that the Borel sets of structures invariant under the subgroup action are exactly the sets of structures definable in Lω1ω(Q).