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The infinite, then, exists in no other way, but in this way it does exist, potentially and by reduction” (Aristotle, Phys. 3. 6, 206b13). According to nearly all interpreters, Aristotle denied both the (actual) existence of infinite magnitude and the (actual) existence of infinite multitude. In the Aristotelian tradition, this apparent denial drove intricate discussions of issues including the past and the status of parts. We’ll look at some consequences for an Aristotelian of denying actual infinities in blanket fashion; then I’ll suggest that Aristotle’s own denial was no blanket one.