Polish group actions and effectivity

Abstract

Abstract. We extend the result of Nadel describing the relationship between approximations of canonical Scott sentences and admissible sets to the general case of orbit equivalence relations induced on an arbitrary Polish space by a Polish group action.Keywords: Polish G-spaces, Scott analysis, recursion, admissible sets Classification: 03E15, 03C70 IntroductionIn the paper we extend the following result of Nadel (see [10]) to the general case of Polish G-spaces.Theorem(Nadel) Let L be a countable language, A be an admissible set and M be an L-structure in A. Then for any L-structure N, if M and N satisfy the same sentences from the admissible fragment L A , then they satisfy the same sentences of quantifier rank α ≤ o(A).The theorem can be formulated in terms of the logic action as follows. Consider a countable relational language L = (R ni i ) i∈I . There is an obvious one-to-one correspondence M → x M between the set of all countable L-structures and the set X L = i∈I 2 ω n i . The set X L equipped with the product topology becomes a Polish space, the space of all L-structures on ω (see Section 2.5 in [4] or Section 2.D of[2] for details). The group S ∞ of all permutations of ω has the natural continuous action on the space X L . It is called the logic action of S ∞ on X L . Given a structure x ∈ X L , the orbit of x under the logic action consists of all structures isomorphic to x. Now the Nadel's theorem reads as follows.