Reflection principle

In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that, with respect to any given property, resemble the class of all sets. There are several different forms of the reflection principle depending on exactly what is meant by "resemble". Weak forms of the reflection principle are theorems of ZF set theory due to , while stronger forms can be new and very powerful axioms for set theory. The name "reflection principle" comes from the fact that properties of the universe of all sets are "reflected" down to a smaller set. A naive version of the reflection principle states that "for any property of the universe of all sets we can find a set with the same property". This leads to an immediate contradiction: the universe of all sets contains all sets, but there is no set with the property that it contains all sets. To get useful (and non-contradictory) reflection principles we need to be more careful about what we mean by "property" and what properties we allow. Reflection principles are associated with attempts to formulate the idea that no one notion, idea, or statement can capture our whole view of the universe of sets. Kurt Gödel described it as follows: The universe of all sets is structurally indefinable. One possible way to make this statement precise is the following: The universe of sets cannot be uniquely characterized (i.e., distinguished from all its initial segments) by any internal structural property of the membership relation in it which is expressible in any logic of finite or transfinite type, including infinitary logics of any cardinal number. This principle may be considered a generalization of the closure principle. All the principles for setting up the axioms of set theory should be reducible to Ackermann's principle: The Absolute is unknowable. The strength of this principle increases as we get stronger and stronger systems of set theory. The other principles are only heuristic principles.