# The Cantor Game: Winning Strategies and Determinacy

## TL;DR

This paper investigates the existence of winning strategies in the Cantor game on uncountable subsets of [0,1], showing that the answers depend on the set theory axioms assumed, such as the Axiom of Determinacy or the Axiom of Choice.

## Contribution

It demonstrates how the existence of winning strategies in the Cantor game varies under different set-theoretic axioms, clarifying the role of these axioms in game determinacy.

## Key findings

- Under the Axiom of Determinacy, no uncountable subset admits a winning strategy for either player.
- Under Zermelo-Fraenkel axioms with the Axiom of Choice, some uncountable subsets allow for a winning strategy for Bob.
- The results depend critically on the set-theoretic assumptions, illustrating the independence of the problem from standard axioms.

## Abstract

In Problem #1542 of Mathematics Magazine, Grossman and Turett define the Cantor game. In his 2007 Mathematics Magazine article about the Cantor game, Matt Baker proves several results and poses three challenging questions about it: Do there exist uncountable subsets of [0, 1] for which: Alice does not have a winning strategy; Bob has a winning strategy; neither Alice nor Bob has a winning strategy? In this paper we show that the answers to these questions depend upon which axioms of set theory are assumed. Specifically, if we assume the Axiom of Determinacy in addition to the Zermelo-Fraenkel axioms, then the answer to all three questions is "no." If instead we assume the Zermelo-Fraenkel axioms together with the Axiom of Choice, then the answer to questions 1 and 3 is "yes," and the answer to question 2 is likely to be "no." Author's Note: This paper was my entry in the 2017 Regeneron Science Talent Search. It earned a Top 300 Scholar Award as well as Research Report and Student Initiative badges.