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combinatorial principle (Definition)

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

$\displaystyle P\implies \Phi\implies Q,$
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, the continuum hypothesis, $ \Diamond$, $ \clubsuit$, and Martin's axiom.

Bibliography

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



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See Also: $\Diamond$, $\clubsuit$

Keywords:  independence, combinatorial, set theory
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Cross-references: Martin's axiom, continuum hypothesis, consistent, consequences, ZF, set theory
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This is version 3 of combinatorial principle, born on 2004-04-10, modified 2005-11-02.
Object id is 5750, canonical name is CombinatorialPrinciple.
Accessed 1450 times total.

Classification:
AMS MSC: 03E65 (Mathematical logic and foundations :: Set theory :: Other hypotheses and axioms)
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