**Erratum -- Martin's conjecture, arithmetic equivalence, and coutable Borel equivalence relations**

Theorem 1.8 (which was stated without proof) is incorrect. While parts (2) and (3) are easily seen to be equivalent, the "easy" induction for showing that (1) and (2) are equivalent fails at the ordinals that are limit stages. In particular, it is open whether the following is true: suppose $f$ is a Turing invariant Borel function such that $f(x) \geq_T x^{(n)}$ on a cone for every $n \in \omega$. Then is it true that $f(x) \geq_T x^{(\omega)}$ on a cone? This is easily seen to be true when $f$ is uniformly Turing invariant. It is also easy to see that $f(x)'' \geq_T x^{(\omega)}$ on a cone (by relativizing the fact that if $y$ is an upper bound for the arithmetic degrees, then $y'' \geq_T 0^{\omega}$).

This error does affect any of the other theorems of the paper.

Thanks to Vittorio Bard for pointing out this error!