Abstract. Let κ < λ be regular uncountable cardinals. Using a finite support iteration of ccc posets we obtain the consistency of b = a = κ < s = λ. If µ is a measurable cardinal and µ < κ < λ, then using similar techniques we obtain the consistency of b = κ < a = s = λ.

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WSPC Proceedings -9in x 6inBrendleBrookeNgNies˙Cichon˙recursion˙theory page 2 2 We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. While this analogy was first studied explicitly by Rupprecht (Effective correspondents to cardinal characteristics in Cichoń's diagram, PhD thesis, University of Michigan, 2010), many prior results can be viewed from this perspective. After a comprehensive survey of the analogy for characteristics from Cichoń's diagram, we extend it to Kurtz randomness and the analogue of the Specker-Eda number.

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For a ⊆ b ⊆ ω with b\a infinite, the set D = {x ∈ [ω] ω : a ⊆ x ⊆ b} is called a doughnut. Doughnuts are equivalent to conditions of Silver forcing, and so, a set S ⊆ [ω] ω is called Silver measurable, also known as completely doughnut, if for every doughnut D there is a doughnut D ⊆ D which is contained or disjoint from S. In this paper, we investigate the Silver measurability of ∆ 1 2 and Σ 1 2 sets of reals and compare it to other regularity properties like the Baire and the Ramsey property and Miller and Sacks measurability.