 NFU + Infinity + Choice is a relatively weak theory: like Zermelo set theory it does not prove even that \(\aleph_{\omega}\) exists. As is the case with Zermelo set theory, natural extensions of this theory make it much stronger. We give just one example. The Axiom of Cantorian Sets is the deceptively simple statement (to which there are no evident counterexamples) that “every cantorian set is strongly cantorian”. NFU + Infinity + Choice + Cantorian Sets is a considerably stronger theory than NFU + Infinity + Choice: in its theory of isomorphism types of well-founded extensional relations with top element, the cantorian types with the obvious “membership” relation satisfy the axioms of ZFC + “there is an \(n\)-Mahlo cardinal” for each concrete \(n\). There is no mathematical need for the devious interpretation: this theory proves the existence of \(n\)-Mahlo cardinals and supports all mathematical constructions at that level of consistency strength in its own terms without any need to refer to the theory of well-founded extensional relations. More elaborate statements about such properties as “cantorian” and “strongly cantorian” (applied to order types as well as cardinality) yield even stronger axioms of infinity.