Abstract

The phrase ‘The iterative conception of sets’ conjures up a picture of a particular settheoretic universe – the cumulative hierarchy – and the constant conjunction of phrasewith-picture is so reliable that people tend to think that the cumulative hierarchy is all there is to the iterative conception of sets: if you conceive sets iteratively, then the result is the cumulative hierarchy. In this paper, I shall be arguing that this is a mistake: the iterative conception of set is a good one, for all the usual reasons. However, the cumulative hierarchy is merely one way among many of working out this conception, and arguments in favour of an iterative conception have been mistaken for arguments in favour of this one special instance of it. (This may be the point to get out of the way the observation that although philosophers of mathematics write of the iterative conception of set, what they really mean – in the terminology of modern computer science at least – is the recursive conception of sets. Nevertheless, having got that quibble off my chest, I shall continue to write of the iterative conception like everyone else.).