combinatorial principle

A combinatorial principle is any statement $\mathrm{\Phi }$ of set theory proved to be independent of Zermelo-Fraenkel (ZF) set theory, usually one with interesting consequences.

If $\mathrm{\Phi }$ is a combinatorial principle, then whenever we have implicationsof the form

$$P⟹\mathrm{\Phi }⟹Q,$$

we automatically know that $P$ is unprovable in ZF and $Q$ is relatively consistent with ZF.

Some examples of combinatorial principles are the axiom of choice (http://planetmath.org/AxiomOfChoice), the continuum hypothesis, $\mathrm{◇}$, $\mathrm{♣}$, and Martin’s axiom.