 Analysis,  or second-order arithmetic, is the extension of first-order arithmetic with the comprehension schema for arbitrary second-order formulae. The theory is impredicative in that it allows one to define sets of natural numbers using quantifiers that range over the entire universe of sets, including, implicitly, the set being defined. One can obtain predicative fragments of this theory by restricting the type of formulae allowed in the comprehension axiom. For example, the restriction discussed in connection with Ackermann above corresponds to the arithmetic comprehension schema, in which formulae do not involve second-order quantifiers. There are various ways of obtaining stronger fragments of analysis that are nonetheless predicatively justified. For example, one obtains ramified analysis by associating an ordinal rank to set variables; roughly, in the definition of a set of a given rank, quantifiers range only over sets of lower rank, i.e., those whose definitions are logically prior.