Degree-invariant, analytic equivalence relations without perfectly many classes

Montalbán, Antonio

doi:10.1090/proc/13218

Degree-invariant, analytic equivalence relations without perfectly many classes
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Proc. Amer. Math. Soc. 145 (2017), 395-398 Request permission

Abstract:

We show that there is only one natural Turing-degree invariant, analytic equivalence relation with $\aleph _1$ many equivalence classes: the equivalence $X \equiv _{\omega _1} Y\iff \omega _1^X=\omega _1^Y$. More precisely, under $PD+\neg CH$, we show that every Turing-degree invariant, analytic equivalence relation with $\aleph _1$ many equivalence classes is equal to $\equiv _{\omega _1}$ on a Turing cone.

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