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.
Read more about this topic: Reflection Principle
Famous quotes containing the words reflection, principles and/or axioms:
“If the contemplation, even of inanimate beauty, is so delightful; if it ravishes the senses, even when the fair form is foreign to us: What must be the effects of moral beauty? And what influence must it have, when it embellishes our own mind, and is the result of our own reflection and industry?”
—David Hume (1711–1776)
“... the history of the race, from infancy through its stages of barbarism, heathenism, civilization, and Christianity, is a process of suffering, as the lower principles of humanity are gradually subjected to the higher.”
—Catherine E. Beecher (1800–1878)
“The axioms of physics translate the laws of ethics. Thus, “the whole is greater than its part;” “reaction is equal to action;” “the smallest weight may be made to lift the greatest, the difference of weight being compensated by time;” and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.”
—Ralph Waldo Emerson (1803–1882)