combinatorial principle

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

If $\Phi$ is a combinatorial principle, then whenever we have implications of the form

 $P\implies\Phi\implies Q,$

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, $\Diamond$, $\clubsuit$, and Martin’s axiom.

References

• 1 Just, W., http://www.math.ohiou.edu/ just/resint.html#principleshttp://www.math.ohiou.edu/~just/resint.html#principles.
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