Reflection Principles As New Axioms
Bernays used a reflection principle as an axiom for one version of set theory (not Gödel-Bernays 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 which has the same property, in other words which is an inaccessible cardinal.) The consistency of Bernays's reflection principle is implied by the existence of a measurable cardinal.
There are many 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 (Marshall R 1989).
If V is a model of ZFC and its class of ordinals is regular, i.e. there is no cofinal subclass of lower order-type, then there is a closed unbounded class of ordinals, C, such that for every αεC, the identity function from Vα to V is an elementary embedding.
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