Reflection principle

In set theory, a branch of mathematics, a reflection principle says that we can find sets that resemble the class of all sets. There are several different forms of the reflection principle depending on exactly one means by "resemble". Weak forms of the reflection principle are theorems of ZF set theory, while stronger forms can be new and very powerful axioms for set theory.

The name "reflection principle" come from the fact that properties of the universe of all sets are "reflected" down to a smaller set.

The reflection principle as a theorem of ZFC

There are several closely related theorems of ZFC all of which state that we can find a set that is almost a model of ZFC. For example, one form of the reflection principle says that for any finite set of axioms of ZFC we can find a countable transitive model satisfying these axioms. (In particular this proves that ZFC is not finitely axiomatizable, because if it were it would prove the existence of a model of itself, and hence prove its own consistency, contradicting Godel's theorem.)

Another version of the reflection principle says that for any finite set of formulas of ZFC we can find a set V_α in the cumulative hierarchy such that all the formulas in the set are absolute for V_α (which means very roughly that they hold in V_α if and only if they hold in the universe of all sets). So this says that the set V_α resembles the universe of all sets, at least as far as the given finite set of formulas is concerned.

References

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