to formulate the idea that no one notion, idea, statement can capture our whole view of the universe of sets. Kurt Gödel described it as follows: Georg Cantor expressed similar views on Absolute Infinity: All cardinality properties are satisfied in this number, in which held by a smaller cardinal. In 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, unless inconsistent, 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 Vα in the cumulative hierarchy such that all the formulas in the set are absolute for Vα (which means very roughly that they hold in Vα if and only if they hold in the universe of all sets). So this says that the set Vα resembles the universe of all sets, at least as far as the given finite number of formulas is concerned. In particular for any formula of ZFC there is a theorem of ZFC that the formula is logically equivalent to a version of it with all quantifiers relativized to Vα. See . If κ is a strong inaccessible cardinal, then there is a closed unbounded subset C of κ, such that for every α∈C, the identity function from Vα to Vκ is an elementary embedding. The reflection principle for ZFC is a theorem schema that can be described as follows: Let be a formula with at most free variables . Then ZFC proves that where denotes the relativization of to (that is, replacing all quantifiers appearing in of the form and by and , respectively). As new axioms Bernays