Elementary submodels, coding strategies, and an infinite real number game

TL;DR

This paper explores the use of set theory and infinite game theory techniques to analyze an infinite-length game related to subsets of real numbers, connecting it to the Banach-Mazur game.

Contribution

It provides new examples of applying set-theoretic and game-theoretic methods to characterize countable subsets of real numbers and links the game to the Banach-Mazur game.

Findings

Demonstrates how set theory techniques answer Baker's question

Connects the game to the Banach-Mazur game in topology

Provides accessible examples of the techniques used

Abstract

Matthew Baker investigated, in previous work, an elegant, infinite-length game that may be used to study subsets of real numbers. We present two accessible examples of how an important technique from set theory, or a different technique from infinite game theory, may be used to answer Baker's question on whether this game provides a precise characterization for countable subsets of real numbers, and we connect this game to the well-studied Banach-Mazur game from topology.