Husserl on Infinity and Idealization

"But with the appearance of Greek Philosophy and its first formulation, through consistent idealization, of the new sense of infinity, there is accomplished in this respect a thoroughgoing transformation [Umwandlung] which finally draws all ideas of finitude and with them all spiritual culture and [its concept of] mankind into its sphere."

Husserl, C 279; K 325.

"Traditionally, Greek philosophy (e.g. Aristotle) did not accept the idea of an actual infinite. The infinitude or 'limitless' (apeiron) as such was incomplete and hence not actual or real."

Modified translation provided by Dermont Moran in Husserl's Crisis of the European Sciences and Transcendental Phenomenology: An Introduction (New York: Cambridge University Press, 2012), p. 87.