Shoenfield's absoluteness theorem shows that formula_1 and formula_2 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 formula_3 sentences are upward absolute (if such a sentence holds in "L" then it holds in "V") and formula_4 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 formula_1 sentences.