Logic Seminar

We show that for every ordinal α∈[1,ω1), there is a closed set F⊂2ω×ωω such that for every x∈2ω, the section {y∈ωω:(x,y)∈F} is a two-point set and F cannot be covered by countably many graphs B(n)⊂2ω×ωω of functions of the variable x∈2ω such that each B(n) is in the additive Borel class Σ0α. This rules out the possibility to have a quantitative version of the Luzin-Novikov theorem. The construction is a modification of the method of Harrington who invented it to show that there exists a countable Π01 set in ωω containing a non-arithmetic singleton. By another application of the same method, we get closed sets excluding a quantitative version of the Saint Raymond theorem on Borel sets with σ-compact sections. (Joint work with P. Holický)