Independence (mathematical logic)

The sentences in this set are referred to as "axioms".

Sometimes, σ is said (synonymously) to be undecidable from T. (This concept is unrelated to the idea of "decidability" as in a decision problem.)

The following statements in set theory are known to be independent of ZF, under the assumption that ZF is consistent: The following statements (none of which have been proved false) cannot be proved in ZFC (the Zermelo–Fraenkel set theory plus the axiom of choice) to be independent of ZFC, under the added hypothesis that ZFC is consistent.

The following statements are inconsistent with the axiom of choice, and therefore with ZFC.

Since 2000, logical independence has become understood as having crucial significance in the foundations of physics.

The parallels axiom ( P ) is independent of the remaining geometry axioms ( R ): there are models (1) that satisfy R and P , but also models (2,3) that satisfy R , but not P .