elements of A which are strongly Cantorian ordinals are exactly the strongly Cantorian ordinals such that φ. Solovay has shown the precise equivalence in consistency strength of NFUB = NFU + Infinity + Cantorian Sets + Small Ordinals with Morse–Kelley set theory plus the assertion that the proper class ordinal (the class of all ordinals) is a weakly compact cardinal. This is very strong indeed! Moreover, NFUB-, which is NFUB with Cantorian Sets omitted, is easily seen to have the same strength as NFUB. A model of the kind constructed above will satisfy this axiom if every collection of ordinals fixed by j is the intersection of some set of ordinals with the ordinals fixed by j, in the underlying nonstandard model of ZFC. Even stronger is the theory NFUM = NFU + Infinity + Large Ordinals + Small Ordinals. This is equivalent to Morse–Kelley set theory with a predicate on the classes which is a κ-complete non-principal ultrafilter on the proper class ordinal κ; in effect, this is Morse–Kelley set theory + "the proper class ordinal is a measurable cardinal"! The technical details here are not the main point, which is that reasonable and natural (in the context of NFU) assertions turn out to be equivalent in power to very strong axioms of infinity in the ZFC context. This fact is related to the correlation between the existence of models of NFU, described above and satisfying these axioms, and the existence of models of ZFC with automorphisms having special properties. See also Alternative set theory Axiomatic set theory Implementation of mathematics in set theory Positive set theory Set-theoretic definition of natural numbers Notes References With discussion by Quine. Quine, W. V., 1980, "New Foundations for Mathematical Logic" in From a Logical Point of View, 2nd ed., revised. Harvard Univ. Press: 80-101. The definitive version of where it all began, namely Quine's 1937 paper in the American Mathematical Monthly. External links "Enriched Stratified systems for the Foundations of Category Theory" by Solomon Feferman (2011) Stanford Encyclopedia of Philosophy: Quine's New Foundations — by Thomas Forster. Alternative axiomatic set theories — by Randall Holmes. Randall Holmes: New Foundations Home Page. Randall Holmes: Bibliography of Set Theory with a Universal Set. Randall Holmes: A new pass at the NF consistency proof Systems of set theory Type theory Urelements Willard Van Orman Quine