[randallholmes]

The project is concerned with NF itself; the status of NFU was settled by Jensen in 1969 (it can be shown to be consistent fairly easily). Showing consistency of NF is difficult.

There is nothing mysterious about urelements: an urelement is simply an object which is not a set. To allow urelements is to weaken extensionality to say, objects with the same elements which are sets are equal; objects which are not sets have no elements (but may be distinct from each other and the empty set).