theorem relates to the axiom of choice. Gödel proved that the constructible universe L always satisfies ZFC, including the axiom of choice, even when V is only assumed to satisfy ZF. Shoenfield's theorem shows that if there is a model of ZF in which a given statement φ is false, then φ is also false in the constructible universe of that model. In contrapositive, this means that if ZFC proves a sentence then that sentence is also provable in ZF. The same argument can be applied to any other principle that always holds in the constructible universe, such as the combinatorial principle ◊. Even if these principles are independent of ZF, each of their consequences is already provable in ZF. In particular, this includes any of their consequences that can be expressed in the (first-order) language of Peano arithmetic. Shoenfield's theorem also shows that there are limits to the independence results that can be obtained by forcing. In particular, any sentence of Peano arithmetic is absolute to transitive models of set theory with the same ordinals. Thus it is not possible to use forcing to change the truth value of arithmetical sentences, as forcing does not change the ordinals of the model to which it is applied. Many famous open problems, such as the Riemann hypothesis and the P = NP problem, can be expressed as sentences (or sentences of lower complexity), and thus cannot be proven independent of ZFC by forcing. Large cardinals There are certain large cardinals that cannot exist in the constructible universe (L) of any model of set theory. Nevertheless, the constructible universe contains all the ordinal numbers that the original model of set theory contains. This "paradox" can be resolved by noting that the defining properties of some large cardinals are not absolute to submodels. One example of such a nonabsolute large cardinal axiom is for measurable cardinals; for an ordinal to be a measurable cardinal there must exist another set (the measure) satisfying certain properties. It can be shown that no such measure is constructible. See also Conservative extension Lévy hierarchy References Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. . Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. . Shoenfield, Joseph, 1961. "The problem of predicativity", Essays on the foundations of mathematics, Y. Bar-Hillel et al., eds., pp. 132–142. Inline citations Mathematical logic Concepts in