The equivalence of determinacy and iterated sharps

We characterize, in terms of determinacy, the existence of 0♯♯ as well as the existence of each of the following: 0♯♯♯, 0♯♯♯♯,0♯♯♯♯♯, ... For k ∈ ω, we define two classes of sets, (k * Σ01)* and (k * Σ01)*+, which lie strictly between $\bigcup_{\beta < \omega^2}(\beta-\Pi^1_1)$ and Δ(ω2-Π11). We also define 01♯ as 0♯ and in general, 0(k + 1)♯ as (0k♯)♯. We then show that the existence of 0(k + 1)♯ is equivalent to the determinacy of ((k + 1) * Σ01)* as well as the determinacy of (k * Σ01)*+.