In this dissertation, I examine Aristotles Physics III exegetically and philosophically to reconstruct how Aristotle discovers the nature of motion as an incomplete actuality, which he uses to argue for an actual infinite. The standard interpretation of Aristotelian infinity is that it exists only potentially because, as Aristotle shows in Phy. III.5, no actual infinite physical body exists whatsoever. However, I contend that because the infinite must exist in order to preserve time, magnitudes, and number, and because any existing thing has some degree of actuality, then the infinite must in some way be actual. The difficulty is figuring out how. I argue that the infinite exists actually as a proximate attribute of motion and primarily as an essential attribute of physical substances. The infinite is an attribute of motion since motion is indefinite. Evidence for this is that in Phy. III.6, Aristotle argues that the infinite is actual like the day and the Olympic games because they are inherently incomplete insofar as they are always in a state of becoming. The day and the games serve as models for how the constant incompleteness of motion is the very phenomenality of infinity. However, as Aristotle argues in Phy. III.1, motion manifests differently across the highest categories of being. This means that not all motions will be strictly the same. Motion is one only by analogy. So, to which motion does the infinite properly belong? Since Aristotle defines the infinite as a sort of quantity, I argue that the infinite belongs to quantitative change, specifically the activities of division and addition. These activities are inherently incomplete, just like the day and the games, whereas all other motions eventually reach completion. But not only is there a peculiar sort of actuality of infinity with respect to motion. I also argue that the actuality of infinity is an essential attribute of physical substances insofar as physical substances undergo division by nature. Substance is the primary source of the actuality of infinity since when actualized in the act of division, albeit incompletely, infinity is always within the actual finite substance.