Paradoxes as a window to infinity

Abstract

This study examines approaches to infinity of two groups of university students with different mathematical background: undergraduate students in Liberal Arts Programmes and graduate students in a Mathematics Education Master's Programme. Our data are drawn from students’ engagement with two well-known paradoxes – Hilbert's Grand Hotel and the Ping-Pong Ball Conundrum – before, during, and after instruction. While graduate students found the resolution of Hilbert's Grand Hotel paradox unproblematic, responses of students in both groups to the Ping-Pong Ball Conundrum were surprisingly similar. Consistent with prior research, the work of participants in our study revealed that they perceive infinity as an ongoing process, rather than a completed one, and fail to notice conflicting ideas. Our contribution is in describing specific challenging features of these paradoxes that might influence students’ understanding of infinity, as well as the persuasive factors in students’ reasoning, that have not been unveiled by other means.

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