to get the effect of strong variants of Infinity; Ali Enayat has investigated the strength of some of these axioms in models of NFU + "the universe is finite". A model of the kind constructed above satisfies Counting just in case the automorphism j fixes all natural numbers in the underlying nonstandard model of Zermelo set theory. The next strong axiom we consider is the Axiom of strongly Cantorian separation: For any strongly Cantorian set A and any formula (not necessarily stratified!) the set exists. Immediate consequences include Mathematical Induction for unstratified conditions (which is not a consequence of Counting; many but not all unstratified instances of induction on the natural numbers follow from Counting). This axiom is surprisingly strong. Unpublished work of Robert Solovay shows that the consistency strength of the theory NFU* = NFU + Counting + Strongly Cantorian Separation is the same as that of Zermelo set theory + Replacement. This axiom holds in a model of the kind constructed above (with Choice) if the ordinals which are fixed by j and dominate only ordinals fixed by j in the underlying nonstandard model of Zermelo set theory are standard, and the power set of any such ordinal in the model is also standard. This condition is sufficient but not necessary. Next is Axiom of Cantorian Sets: Every Cantorian set is strongly Cantorian. This very simple assertion is extremely strong. Solovay has shown the precise equivalence of the consistency strength of the theory NFUA = NFU + Infinity + Cantorian Sets with that of ZFC + a schema asserting the existence of an n-Mahlo cardinal for each concrete natural number n. Ali Enayat has shown that the theory of Cantorian equivalence classes of well-founded extensional relations (which gives a natural picture of an initial segment of the cumulative hierarchy of ZFC) interprets the extension of ZFC with n-Mahlo cardinals directly. A permutation technique can be applied to a model of this theory to give a model in which the hereditarily strongly Cantorian sets with the usual membership relation model the strong extension of ZFC. This axiom holds in a model of the kind constructed above (with Choice) just in case the ordinals fixed by j in the underlying nonstandard model of ZFC are an initial (proper class) segment of the ordinals of the model. Next consider the Axiom of Cantorian Separation: For any Cantorian set A and any