# 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 implications^{}
of the form

$$P\u27f9\mathrm{\Phi}\u27f9Q,$$ |

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^{}, $\mathrm{\u25c7}$, $\mathrm{\u2663}$, 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.

Title | combinatorial principle |
---|---|

Canonical name | CombinatorialPrinciple |

Date of creation | 2013-03-22 14:17:41 |

Last modified on | 2013-03-22 14:17:41 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 6 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 03E65 |

Related topic | Diamond^{} |

Related topic | Clubsuit |