On the Infinite in Aristotle

Abstract

In this paper we state several propositions concerning the infinite in Aristotle and some remarks relating to them.
Preliminaries. (a) Usually περαç or απειρον is translated into limit or infinite respectively, but boundary or non-bounded seems more suitable. If we aim at exactness, we must define all notions such as finite, infinite, boundary and non-bounded in a set-theoretical sense. But in this paper we use these notions without such definitions. (b) The infinite is the privation of a boundary, and, according to the sense of privation, the law of the excluded middle holds between the infinite and the finite. The following propositions hold in Aristotle.
Prop. 1. As the infinite exists potentially, it cannot exist actually and it cannot be a whole. So the whole of all natural numbers can-not exist.
Prop. 2. As the finite exists actually and the infinite exists potentially, the former is prior to the latter. In other words, the finite and the infinite are not coordinate with respect to rank of being.
Prop. 3. The continuous can be divided ad infinitum, and in this sense it contains the infinite in itself. On the other hand, the infinite surpasses the finite, and in this sense the former contains the latter in itself. Consequently, the inifinite is between the finite and the continuous.
Prop. 4. The infinite can be generated in the movement, and, according to the sense of generation, the infinite is between being and non-being.
Remark 1. Georg Cantor defined a set as jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unsrer Anschauung oder unseres Denkens welche die Elemente von M genannt werden zu einem Ganzen. If we interpret das Ganze of Cantor as a whole of Aristotle, a paradox will arise in set theory. For, while a whole must be finite or bounded, an infinite or non-bounded set, for example, a set of all natural numbers exists in set theory.
Remark 2. In virtue of propositions 1 and 2, Aristotle differs from a finite standpoint, which is represented by D. Hilbert and G. Gentzenin the foundations of mathematics, and which allows the use of transfinite induction.