In general, the infinite exists through one thing being taken after another, what is taken being always finite, but ever other and other.
Aristotle (384 BC-322 BC) is a touch-stone for this whole enquiry. Born in Stagira, he lived most of his life in Athens, where he studied under Plato in the Academy that he (Plato) had founded. He was a remarkable polymath. He made major contributions to logic, metaphysics, the natural sciences (above all biology), psychology, ethics, politics, and literary criticism; and some of these disciplines he can even be said to have founded. His entry into our particular drama in many respects marks the end of the prologue and serves to inaugurate the action proper. Many of the concepts that have shaped and informed subsequent discussion, indeed much of what has actually been discussed, originated with Aristotle.
He himself began with the views of his predecessors. He noted in particular one recurring and dominant theme in what they had been saying: whatever is infinite is ipso facto a ‘principle’, that is to say something fundamental from which other things are derived or in terms of which other things are to be explained. Otherwise, it would be derivative and thus limited. This is why, despite profound differences in their views, earlier thinkers had all held the infinite to be ungenerable, indestructible, and eternal. It is also why they had tended to refrain from attributing any particular or determinate qualities to it. These too would have counted as limiting it, leaving it open to explanation in more fundamental terms.
Such remarks apply both to those thinkers who regarded the infinite as an entity in its own right (paradigmatically Anaximander, for whom it was actually a substance) and to those who thought of infinitude in a more modern vein as a property that other entities possessed. Among the latter were some philosophers not discussed above. Anaxagoras, for example, had held that all the different substances in the world were originally