Degree-invariant, analytic equivalence relations without perfectly many classes
Antonio Montalbán · Proceedings of the American Mathematical Society · 2016
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 $PD+ eg CH$, 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.