Axiom of determinacy for natural-number games

Using Baire sequence space ${\mathbb{N}}^{\mathbb{N}}$ and its cylinder topology and the full-position strategy convention of Gale–Stewart games and strategies, the Axiom of Determinacy (AD) is the assertion

$(\mathrm{\forall }A\subseteq {\mathbb{N}}^{\mathbb{N}})G(A;{\mathbb{N}}^{<\omega })\text{ is determined}\mathrm{.}$

Every position allows every natural-number move. Player I moves first. The axiom concerns all payoff sets on this one countable alphabet, including the empty payoff and the whole space; it is not an assertion of determinacy for games on arbitrary sets of moves. For the empty payoff the constant-zero II strategy wins; for the whole-space payoff the constant-zero I strategy wins, directly from the winning condition. The definition itself does not assume AD. Any theorem using it states the assumption. In particular this definition neither asserts unrestricted dependent choice nor asserts compatibility with AC.