Long games just beyond fixed countable length

TL;DR

This paper introduces a new class of variable-length games on natural numbers, linking their determinacy to the existence of a specific inner model with a limit of Woodin cardinals, extending previous work on fixed-length games.

Contribution

It defines a novel type of infinite game with variable countable length and establishes its determinacy equivalence to a significant inner model with Woodin cardinals, building on prior research.

Findings

Analytic determinacy is equivalent to the existence of a sharp for a canonical inner model.

The inner model has a limit of Woodin cardinals with a specific order type.

The work extends the understanding of game determinacy beyond fixed-length scenarios.

Abstract

We introduce a new type of game on natural numbers of variable countable length, which can be regarded as a diagonalization of all games of fixed countable length on natural numbers. Building on previous work by Trang and Woodin, we show that analytic determinacy of the game is equivalent to the existence of a sharp for a canonical inner model with a limit of Woodin cardinals $\lambda$ such that the order type of Woodin cardinals below $\lambda$ is $\lambda$.