structural:scholar_edition, linguistic:translation
Independence

Many important statements are independent of ZFC. The independence is usually proved by forcing, whereby it is shown that every countable transitive model of ZFC (sometimes augmented with large cardinal axioms) can be expanded to satisfy the statement in question. A different expansion is then shown to satisfy the negation of the statement. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particular inner models, such as in the constructible universe. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms.

Forcing proves that the following statements are independent of ZFC:

Remarks:

A variation on the method of forcing can also be used to demonstrate the consistency and unprovability of the axiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice. (Thus every model of ZF contains a submodel of ZFC, so that Con(ZF) implies Con(ZFC).) Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one satisfying ZF but not C.

Another method of proving independence results, one owing nothing to forcing, is based on Gödel's second incompleteness theorem. This approach employs the statement whose independence is being examined, to prove the existence of a set model of ZFC, in which case Con(ZFC) is true. Since ZFC satisfies the conditions of Gödel's second theorem, the consistency of ZFC is unprovable in ZFC (provided that ZFC is, in fact, consistent). Hence no statement allowing such a proof can be proved in ZFC. This method can prove that the existence of large cardinals is not provable in ZFC, but cannot prove that assuming such cardinals, given ZFC, is free of contradiction.