Some results on projective equivalence relations

[摘要]

We construct a ∏¹₁ equivalence relation E on ω^ω for which there is no largestE-thin, E-invariant ∏¹₁ subset of ω^ω. Then we lift our result to the general case.Namely, we show that there is a ∏¹_2n+1 equivalence relation for which there isno largest E-thin, E-invariant ∏¹_2n+1 set under projective determinacy. Thisanswers an open problem raised in Kechris [Ke2].

Our second result in this thesis is a representation for thin ∏¹₃ equivalencerelations on u_ω. Precisely, we show that for each thin ∏¹₃ equivalence relationE on u_ω, there is a Δ¹₃ in the codes map p: ω^ω → u_ω and a ∏¹₃ in the codesequivalence relation e on u_ω such that for all real numbers x and y,

xEy ↔ (p(x),p(y))∈ e

This lifts Harrington's result about thin ∏¹₁ equivalence relations to thin ∏¹₃equivalence relations.

[发布机构] University:California Institute of Technology;Department:Physics, Mathematics and Astronomy

[关键词] Mathematics