The ABCD hypothesis

If α and β are cardinals, then the cardinal exponentiation α β is the set of all functions from α into β . The ABCD hypothesis is the statement that for all infinite cardinals α ≤ β and γ ≤ δ , | α β | ≤ | γ δ | if and only if α ≤ γ and β ≤ δ . The Axiom of Determinacy, AD, states that every two-player infinite game on ω has a winning strategy for one of the two players. Let Θ be the supremum of the ordinals onto which R surjects. This paper will show that if AD holds and V = L ( R ) , then the ABCD hypothesis holds below Θ .