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Matthew Baker investigated, in previous work, an elegant, infinite-lengthgame that may be used to study subsets of real numbers. We present twoaccessible examples of how an important technique from set theory, or adifferent technique from infinite game theory, may be used to answer Baker'squestion on whether this game provides a precise characterization for countablesubsets of real numbers, and we connect this game to the well-studiedBanach-Mazur game from topology.

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