fixed model of set theory and only consider other transitive models containing the same ordinals as the fixed model. Certain properties are absolute to all transitive models of set theory, including the following (see Jech (2003 sec. I.12) and Kunen (1980 sec. IV.3)). x is the empty set. x is an ordinal. x is a finite ordinal. x is a successor ordinal. x is a limit ordinal. x = ω. x is (the graph of) a function. Other properties are not absolute: being finite being countable being a cardinal being a regular cardinal being a limit cardinal being an inaccessible cardinal Failure of absoluteness for countability Skolem's paradox is the seeming contradiction that on the one hand, the set of real numbers is uncountable (and this is provable from ZFC, or even from a small finite subsystem ZFC' of ZFC), while on the other hand there are countable transitive models of ZFC' (this is provable in ZFC), and the set of real numbers in such a model will be a countable set. The paradox can be resolved by noting that countability is not absolute to submodels of a particular model of ZFC. It is possible that a set X is countable in a model of set theory but uncountable in a submodel containing X, because the submodel may contain no bijection between X and ω, while the definition of countability is the existence of such a bijection. The Löwenheim–Skolem theorem, when applied to ZFC, shows that this situation does occur. Shoenfield's absoluteness theorem Shoenfield's absoluteness theorem shows that and sentences in the analytical hierarchy are absolute between a model V of ZF and the constructible universe L of the model, when interpreted as statements about the natural numbers in each model. The theorem can be relativized to allow the sentence to use sets of natural numbers from V as parameters, in which case L must be replaced by the smallest submodel containing those parameters and all the ordinals. The theorem has corollaries that sentences are upward absolute (if such a sentence holds in L then it holds in V) and sentences are downward absolute (if they hold in V then they hold in L). Because any two transitive models of set theory with the same ordinals have the same constructible universe, Shoenfield's theorem shows that two such models must agree about the truth of all sentences. One consequence of Shoenfield's