technical details to verify, but there is an interpretation not only of a fragment of ZFC but of NFU itself in this structure, with defined as : this "relation" is not a set relation but has the same type displacement between its arguments as the usual membership relation . So there is a natural construction inside NFU of the cumulative hierarchy of sets which internalizes the natural construction of a model of NFU in Zermelo-style set theory. Under the Axiom of Cantorian Sets described in the New Foundations article, the strongly cantorian part of the set of isomorphism classes of set pictures with the E relation as membership becomes a (proper class) model of ZFC (in which there are n-Mahlo cardinals for each n; this extension of NFU is strictly stronger than ZFC). This is a proper class model because the strongly cantorian isomorphism classes do not make up a set. Permutation methods can be used to create from any model of NFU a model in which every strongly cantorian isomorphism type of set pictures is actually realized as the restriction of the true membership relation to the transitive closure of a set. See also Axiomatic set theory References Keith Devlin, 1994. The Joy of Sets, 2nd ed. Springer-Verlag. Holmes, Randall, 1998. Elementary Set Theory with a Universal Set. Academia-Bruylant. The publisher has graciously consented to permit diffusion of this introduction to NFU via the web. Copyright is reserved. Potter, Michael, 2004. Set Theory and its Philosophy, 2nd ed. Oxford Univ. Press. Suppes, Patrick, 1972. Axiomatic Set Theory. Dover. Tourlakis, George, 2003. Lectures in Logic and Set Theory, Vol. 2. Cambridge Univ. Press. External links Metamath: A web site devoted to an ongoing derivation of mathematics from the axioms of ZFC and first-order logic. Stanford Encyclopedia of Philosophy: Quine's New Foundations—by Thomas Forster. Alternative axiomatic set theories—by Randall Holmes. Randall Holmes: New Foundations Home Page Large-scale mathematical formalization projects Formalism (deductive) Mathematical logic Set theory