Total Functionals and Well-Founded Strategies

In existing game models, total functionals have no simple characterization neither in term of game strategies, nor in term of the total set-theoretical functionals they de ne. We show that the situation changes if we extend the usual notion of game by allowing in nite plays. Total functionals are, now, exactly those having a tree-strategy in which all branches end in a last move, winning for the strategy. Total functionals now de ne (via an extensional collapse) all set-theoretical functionals. Our model is concrete: we used in nite computations only to have a nice characterization of totality. A computation may be in nite only when the input is a discontinous functional; in practice, never.