Wittgenstein's conception of infinity can be seen as continuing the tradition of the potential infinite that begins with Aristotle. Transfinite cardinals in set theory might seem to render the potential infinite defunct with the actual infinite now given mathematical legitimacy. But Wittgenstein's remarks on set theory argue that the philosophical notion of the actual infinite remains philosophical and is not given a mathematical status as a result of set theory. The philosophical notion of the actual infinite is not to be (...) found in the mathematics of set theory, only in a certain associated philosophy – what Wittgenstein calls a certain kind of “prose”. (shrink)