A parametrised choice principle and Martin's conjecture on Blackwell determinacy

Tony Martin proved in [Mar98] that the Axiom of Determinacy AD implies the Axiom of Blackwell Determinacy Bl-AD, and conjectured that the converse holds as well: Conjecture 1 The Axiom of Blackwell Determinacy Bl-AD implies the Axiom of Determinacy AD. Although Martin, Neeman and Vervoort in their [MarNeeVer03] have made considerable progress on Martin’s Conjecture 1, the full conjecture remains open. In this note, we investigate a fragment of the Axiom of Choice that we shall call the Parametrised Choice Principle PCP which captures the difference between the two axioms (if there is one). Our result is that (in the base theory ZF+Bl-AD), AD and PCP are equivalent (Theorem 3). So, showing Martin’s Conjecture 1 is tantamount to proving the choice principle from Bl-AD.