Glimpses of infinity: intuitions, paradoxes, and cognitive leaps

This dissertation examines undergraduate and graduate university students’ emergent conceptions of mathematical infinity. In particular, my research focuses on identifying the cognitive leaps required to overcome epistemological obstacles related to the idea of actual infinity. Extending on prior research regarding intuitive approaches to set comparison tasks, my research offers a refined analysis of the tacit conceptions and philosophies which influence learners’ emergent understanding of mathematical infinity, as manifested through their engagement with geometric tasks and two well known paradoxes – Hilbert’s Grand Hotel paradox and the Ping-Pong Ball Conundrum. In addition, my research sheds new light on specific features involved in accommodating the idea of actual infinity. The results of my research indicate that accommodating the idea of actual infinity requires a leap of imagination away from ‘realistic’ considerations and philosophical beliefs towards the ‘realm of mathematics’. The abilities to clarify a separation between an intuitive and a formal understanding of infinity, and to conceive of ‘infinite’ as an answer to the question ‘how many?’ are also recognised as fundamental features in developing a normative understanding of actual infinity. Further, in order to accommodate the idea of actual infinity it is necessary to understand specific properties of transfinite arithmetic, in particular the indeterminacy of transfinite subtraction.