Abstract
What would mathematics lose if completed infinite collections were excluded? This expository essay follows the question from Aristotle’s potential infinity through Dedekind’s real numbers and Cantor’s hierarchy of infinite cardinalities to contemporary finite and constructive objections. It distinguishes claims about mathematical existence from claims about physical reality, and the removal of completed infinite objects from a fixed ceiling on finite reasoning. Examples from limits, diagonal arguments, and the foundations of analysis show both the explanatory power of classical infinity and the legitimacy of asking how much infinity particular results require. The essay defends infinity’s central role in classical mathematical practice while recognizing that useful finite computations and alternative foundations do not depend on accepting every classical infinite structure. Its purpose is a historical and philosophical synthesis for a broad mathematical readership.
This page presents the abstract. The full manuscript is not posted here.