Reflection Principle - The Reflection Principle As A Theorem of ZFC

The Reflection Principle As A Theorem of ZFC

In trying to formalize the argument for the reflection principle of the previous section in ZF set theory, it turns out to be necessary to add some conditions about the collection of properties A (for example, A might be finite). Doing this produces several closely related "reflection theorems" of ZFC all of which state that we can find a set that is almost a model of ZFC.

One form of the reflection principle in ZFC 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 Gödel's second incompleteness theorem.) This version of the reflection theorem is closely related to the Löwenheim-Skolem theorem.

Another version of the reflection principle says that for any finite number 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 number of formulas is concerned.

For any natural number n, one can prove from ZFC a reflection principle which says that given any ordinal α, there is an ordinal β>α such V_β satisfies all first order sentences of set theory which are true for V and contain fewer than n quantifiers.