Turing degree
formula_18 (lightface) determinacy is equiconsistent with a Woodin cardinal. If formula_18 determinacy holds, then for a Turing cone of "x" (that is for every real "x" of sufficiently high Turing degree), L["x"] satisfies OD-determinacy (that is determinacy of games on integers of length ω and ordinal-definable payoff), and in HODL["x"] formula_20 is a Woodin cardinal.