Completing the infinite
Anthony writes "An infinite procedure is, by definition, a procedure which can never be completed". Well no. By definition, an infinite procedure is one that can never be completed in a finite amount of time. To say it can never be completed at all begs the question, for it presumes that all processes are finite.
Cantor: If one considers the arguments which Aristotle presented against the real existence of the infinite (vid. his Metaphysics, Book XI, Chap. 10), it will be found that they refer back to an assumption, which involves a petitio principii, the assumption, namely, that there are only finite numbers, from which he concluded that to him only enumerations of finite sets were recognizable. [Punktmannigfaltigkeiten § 4 p. 104-5]You ask what 'finite' means, which Russell answers as follows.
§ 183 Of all the philosophers who have inveighed against infinite number, I doubt whether there is one who has known the difference between finite and infinite numbers. The difference is simply this. Finite numbers obey the law of mathematical induction; infinite numbers do not… It is in this alone, and in its consequences, that finite and infinite numbers differ.
The principle may otherwise be stated thus: If every proposition which holds concerning 0, and also holds concerning the immediate successor of every number of which it holds, holds concerning the number n, then n is finite; if not, not. This is the precise sense of what may be popularly expressed by saying that every finite number can be reached from 0 by successive steps, or by successive additions of 1. This the principle which the philosopher must be held to lay down as obviously applicable to all numbers, though he will have to admit that the more precisely his principle is stated, the less obvious it becomes. [The Principles of Mathematics]