Abstract A Cantor series expansion for a real number x with respect to a basic sequence $Q=(q_1,q_2,\dots )$ , where $q_i \geq 2$ , is a generalization of the base b expansion to an infinite sequence of bases. Ki and Linton in 1994 showed that for ordinary base b expansions the set of normal numbers is a $\boldsymbol {\Pi }^0_3$ -complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio normality, and distribution normality. These notions are equivalent for base b expansions, but not for more general Cantor series expansions. We show that for any basic sequence the set of distribution normal numbers is $\boldsymbol {\Pi }^0_3$ -complete, and if Q is $1$ -divergent then the sets of normal and ratio normal numbers are $\boldsymbol {\Pi }^0_3$ -complete. We further show that all five non-trivial differences of these sets are $D_2(\boldsymbol {\Pi }^0_3)$ -complete if $\lim _i q_i=\infty $ and Q is $1$ -divergent. This shows that except for the trivial containment that every normal number is ratio normal, these three notions are as independent as possible.

Abstract We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain features.

Abstract Assuming AD, we show that all of the ordinals below $\delta _5^1$ represented by descriptions (c.f. [2], but also defined below) are cardinals. Using this analysis we also get a simple representation for the cardinal structure below $\delta _5^1$ . As an application, we compute the cofinalitites of all cardinals below $\delta _5^1$ .

Abstract We work under the assumption of the Axiom of Determinacy and associate a measure to each cardinal κ < ℵ ε0 in a recursive definition of a canonical measure assignment. We give algorithmic applications of the existence of such a canonical measure assignment (computation of cofinalities, computation of the Kleinberg sequences associated to the normal ultrafilters on all projective ordinals).

Abstract Let E be a coanalytic equivalence relation on a Polish space X and (An)n∈ω a sequence of analytic subsets of X. We prove that if lim supn∈kAn meets uncountably many E-equivalence classes for every K ∈ [ω]ω, then there exists K ∈ [ω]ω such that ∩n∈kAn contains a perfect set of pairwise E-inequivalent elements.

Abstract We show that all the projective ordinals are supercompact through their supremum and a ways beyond.

Abstract We formulate and prove a combinatorial property assuming AD + V = L(ℝ). As a consequence, we show that every regular κ which is either a Suslin cardinal or the successor of a Suslin cardinal is -supercompact. In particular, all the projective ordinals are -supercompact.

Abstract Assuming AD + (V = L(R)), it is shown that for κ an admissible Suslin cardinal, o(κ) (= the order type of the stationary subsets of κ) is “essentially” regular and closed under ultrapowers in a manner to be made precise. In particular, o(κ) ≫ κ+, κ++, etc. It is conjectured that this characterizes admissibility for L(R).

Abstract Let X and Y be uncountable Polish spaces. We show in ZF that there is a coanalytic subset P of X × Y with countable sections which cannot be expressed as the union of countably many partial coanalytic, or even PCA = , graphs. If X = Y = ωω, P may be taken to be . Assuming stronger set theoretic axioms, we identify the least pointclass such that any such coanalytic P can be expressed as the union of countably many graphs in this pointclass. This last result is extended (under suitable hypotheses) to all levels of the projective hierarchy.