class theory Paul Bernays used a reflection principle as an axiom for one version of set theory (not Von Neumann–Bernays–Gödel set theory, which is a weaker theory). His reflection principle stated roughly that if A is a class with some property, then one can find a transitive set u such that A∩u has the same property when considered as a subset of the "universe" u. This is quite a powerful axiom and implies the existence of several of the smaller large cardinals, such as inaccessible cardinals. (Roughly speaking, the class of all ordinals in ZFC is an inaccessible cardinal apart from the fact that it is not a set, and the reflection principle can then be used to show that there is a set that has the same property, in other words that is an inaccessible cardinal.) Unfortunately, this cannot be axiomatized directly in ZFC, and a class theory like Morse–Kelley set theory normally has to be used. The consistency of Bernays's reflection principle is implied by the existence of an ω-Erdős cardinal. More precisely, the axioms of Bernays' class theory are: extensionality class specification: for any formula without free, subsets: reflection: for any formula , foundation choice where denotes the powerset. According to Akihiro Kanamori, in a 1961 paper, Bernays considered the reflection schema for any formula without free, where asserts that is transitive. Starting with the observation that set parameters can appear in and can be required to contain them by introducing clauses into , Bernays just with this schema established pairing, union, infinity, and replacement, in effect achieving a remarkably economical presentation of ZF. Others Some formulations of Ackermann set theory use a reflection principle. Ackermann's axiom states that, for any formula not mentioning , Peter Koellner showed that a general class of reflection principles deemed "intrinsically justified" are either inconsistent or weak, in that they are consistent relative to the Erdös cardinal. However, there are more powerful reflection principles, which are closely related to the various large cardinal axioms. For almost every known large cardinal axiom there is a known reflection principle that implies it, and conversely all but the most powerful known reflection principles are implied by known large cardinal axioms. An example of this is the wholeness axiom, which implies the existence of super-n-huge cardinals for all finite n and its consistency is implied by an I3 rank-into-rank cardinal. Add an axiom saying that