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 ${\displaystyle \Pi _{2}^{1}}$ and ${\displaystyle \Sigma _{2}^{1}}$ 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 ${\displaystyle \Sigma _{3}^{1}}$ sentences are upward absolute (if such a sentence holds in L then it holds in V)^[1] and ${\displaystyle \Pi _{3}^{1}}$ 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 ${\displaystyle \Pi _{2}^{1}}$ sentences.

One consequence of Shoenfield's 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 ${\displaystyle \Sigma _{3}^{1}}$ 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 ${\displaystyle \Sigma _{3}^{1}}$ 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 ${\displaystyle \Sigma _{3}^{1}}$ 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 ${\displaystyle \Pi _{2}^{1}}$ 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.

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