  The conception of set underlying NF arises from a related – indeed complementary – oversimplification. ZF arises from the belief that if we enforce a kind of wellfoundedness in the constitution of set then all will be well. NF arises ultimately from the idea that all will be well if we enforce a kind of wellfoundedness (or at least noncircularity) in the description of sets. This concern can take two forms: One might feel uneasy about conjuring up a set by means of a spell that invokes other sets that are in some sense “not yet” available (sets of which the candidate will turn out to be a member, for example). This is a mathematically fruitful concern (keyword “predicativity”) and it will reappear later in the study of NF (and in ZF too, of course) but it is not this first form that gives rise to the axiomatisation of NF. In the second view, the key to neutralising the dangers of self-reference lies in segregating sets into pairwise disjoint levels called types. (That this concern is orthogonal to the first kind of concern can be shown by consideration of the axiom of sumset. This is a legitimate axiom from the second point of view, even though the spell that conjures \(\bigcup X\) does it by invoking the set \(X\), which is of higher type.)