Reflection Principle - Motivation For Reflection Principles

Motivation For Reflection Principles

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.

To find non-contradictory reflection principles we might argue informally as follows. Suppose that we have some collection A of methods for forming sets (for example, taking powersets, subsets, the axiom of replacement, and so on). We can imagine taking all sets obtained by repeatedly applying all these methods, and form these sets into a class V, which can be thought of as a model of some set theory. But now we can introduce the following new principle for forming sets: "the collection of all sets obtained from some set by repeatedly applying all methods in the collection A is also a set". If we allow this new principle for forming sets, we can now continue past V, and consider the class W of all sets formed using the principles A and the new principle. In this class W, V is just a set, closed under all the set-forming operations of A. In other words the universe W contains a set V which resembles W in that it is closed under all the methods A.

We can use this informal argument in two ways. We can try to formalize it in (say) ZF set theory; by doing this we obtain some theorems of ZF set theory, called reflection theorems. Alternatively we can use this argument to motivate introducing new axioms for set theory.