This paper discusses a sequence of extensions of NFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of .
The original theory NF of Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in  that NF disproves Choice (and so proves Infinity). Even if one assumes the consistency of NF, one is hampered by the lack of powerful methods for proofs of consistency and independence such as are available for use with ZFC; very clever work has been done with permutation methods, starting with  and , and exemplified more recently by , but permutation methods can only be applied to show the consistency or independence of unstratified sentences (see the definition of NFU below for a definition of stratification). For example, there is no method available to determine whether the assertion “the continuum can be well-ordered” is consistent with or independent of NF. There is one substantial independence result for an assertion with nontrivial stratified consequences, using metamathematical methods: this is Orey's proof of the independence of the Axiom of Counting from NF (see below for a statement of this axiom).
We mention these difficulties only to reassure the reader of their irrelevance to the present work. Jensen's modification of “New Foundations” (in ), which was to restrict extensionality to sets, allowing many non-sets (urelements) with no elements, has almost magical effects.