formula (not necessarily stratified!) the set {x∈A|φ} exists. This combines the effect of the two preceding axioms and is actually even stronger (precisely how is not known). Unstratified mathematical induction enables proving that there are n-Mahlo cardinals for every n, given Cantorian Sets, which gives an extension of ZFC that is even stronger than the previous one, which only asserts that there are n-Mahlos for each concrete natural number (leaving open the possibility of nonstandard counterexamples). This axiom will hold in a model of the kind described above if every ordinal fixed by j is standard, and every power set of an ordinal fixed by j is also standard in the underlying model of ZFC. Again, this condition is sufficient but not necessary. An ordinal is said to be Cantorian if it is fixed by T, and strongly Cantorian if it dominates only Cantorian ordinals (this implies that it is itself Cantorian). In models of the kind constructed above, Cantorian ordinals of NFU correspond to ordinals fixed by j (they are not the same objects because different definitions of ordinal numbers are used in the two theories). Equal in strength to Cantorian Sets is the Axiom of Large Ordinals: For each non-Cantorian ordinal , there is a natural number n such that . Recall that is the order type of the natural order on all ordinals. This only implies Cantorian Sets if we have Choice (but is at that level of consistency strength in any case). It is remarkable that one can even define : this is the nth term of any finite sequence of ordinals s of length n such that , for each appropriate i. This definition is completely unstratified. The uniqueness of can be proved (for those n for which it exists) and a certain amount of common-sense reasoning about this notion can be carried out, enough to show that Large Ordinals implies Cantorian Sets in the presence of Choice. In spite of the knotty formal statement of this axiom, it is a very natural assumption, amounting to making the action of T on the ordinals as simple as possible. A model of the kind constructed above will satisfy Large Ordinals, if the ordinals moved by j are exactly the ordinals which dominate some in the underlying nonstandard model of ZFC. Axiom of Small Ordinals: For any formula φ, there is a set A such that the