but it is more natural to consider assertions about Cantorian and strongly Cantorian sets. Such assertions not only bring into being large cardinals of the usual sorts, but strengthen the theory on its own terms. The weakest of the usual strong principles is: Rosser's Axiom of Counting. The set of natural numbers is a strongly Cantorian set. To see how natural numbers are defined in NFU, see set-theoretic definition of natural numbers. The original form of this axiom given by Rosser was "the set {m|1≤m≤n} has n members", for each natural number n. This intuitively obvious assertion is unstratified: what is provable in NFU is "the set {m|1≤m≤n} has members" (where the T operation on cardinals is defined by ; this raises the type of a cardinal by one). For any cardinal number (including natural numbers) to assert is equivalent to asserting that the sets A of that cardinality are Cantorian (by a usual abuse of language, we refer to such cardinals as "Cantorian cardinals"). It is straightforward to show that the assertion that each natural number is Cantorian is equivalent to the assertion that the set of all natural numbers is strongly Cantorian. Counting is consistent with NFU, but increases its consistency strength noticeably; not, as one would expect, in the area of arithmetic, but in higher set theory. NFU + Infinity proves that each exists, but not that exists; NFU + Counting (easily) proves Infinity, and further proves the existence of for each n, but not the existence of . (See beth numbers). Counting implies immediately that one does not need to assign types to variables restricted to the set of natural numbers for purposes of stratification; it is a theorem that the power set of a strongly Cantorian set is strongly Cantorian, so it is further not necessary to assign types to variables restricted to any iterated power set of the natural numbers, or to such familiar sets as the set of real numbers, the set of functions from reals to reals, and so forth. The set-theoretical strength of Counting is less important in practice than the convenience of not having to annotate variables known to have natural number values (or related kinds of values) with singleton brackets, or to apply the T operation in order to get stratified set definitions. Counting implies Infinity; each of the axioms below needs to be adjoined to NFU + Infinity