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We show that there is only one natural Turing-degree invariant, analytic equivalence relation with א1 many equivalence classes: the equivalence X ≡ω1 Y ⇐⇒ ω 1 = ω Y 1 . More precisely, under PD + ¬CH, we show that every Turing-degree invariant, analytic equivalence relation with א1 many equivalence classes is equal to ≡ω1 on a Turing-cone.

Degree-invariant, analytic equivalence relations without perfectly many classes