As Charlton (1991, p.129) notes, to understand a thinker‟s finitism it is important to understand what kinds of infinity she rejects. Moreover, as I shall argue, to understand Aristotle‟s finitism, it is central to understand not only what kinds of infinity he rejects but also on what grounds he rejects them, for that will allow us to recognize some subtleties of Aristotle‟s account that would otherwise remain unnoticed. In this paper, I attempt to give a systematic account of Aristotle‟s notion of the potentially infinite by closely scrutinizing Physics III 6. In part 1, I discuss Aristotle‟s argument that the infinite exist only potentially. In part 2, I argue that the distinction between infinite processes and the products of these processes is also relevant for understanding Aristotle‟s views on the existence of the infinite.