Axiom independence
An axiom P is independent of a theory T if you cannot prove P from T and you cannot prove not P from T. In other words, there are models of T where P is true and models of T where P is false. Independence is usually shown by exhibiting such models. If T is inconsistent (you can prove every sentence from it), then no axiom is independent. Proving independence can be hard, but forcing is a common method, used to show that the continuum hypothesis is independent of ZFC.
Independence matters because it helps us find the smallest set of axioms that are actually needed, and it plays a key role in reverse mathematics, which asks which axioms are necessary to prove a theorem.
A classic example is the parallel postulate. It is independent from the first four postulates, so you can have geometries where it holds and others where it does not. Beltrami showed that hyperbolic geometry, which allows more than one parallel, is consistent with the first four postulates. This uses models such as the Beltrami-Klein model, the Poincaré disk, and the Poincaré half-plane. Playfair’s axiom is a convenient way to state the parallel postulate: through a given line and a point not on it, there is at most one parallel line.
Euclidean geometry satisfies all five postulates; hyperbolic geometry satisfies the first four plus at least two distinct parallels; elliptic (spherical) geometry has no parallels.
The continuum hypothesis (CH) asks whether there is a set whose size lies strictly between the integers and the real numbers. Gödel showed CH cannot be disproved from ZF (assuming ZF is consistent) by showing CH holds in the constructible universe L. Cohen later showed CH cannot be proven from ZFC, using the forcing method. This established that CH is independent of ZFC, a result for which Cohen received the Fields Medal in 1966.
This page was last edited on 2 February 2026, at 02:52 (CET).