Editing Independence (mathematical logic) (section)

==Independence results in set theory==

Many interesting statements in set theory are independent of [[Zermelo–Fraenkel set theory]] (ZF). The following statements in set theory are known to be independent of ZF, under the assumption that ZF is consistent:
*The [[axiom of choice]]
*The [[continuum hypothesis]] and the [[Continuum hypothesis#The generalized continuum hypothesis|generalized continuum hypothesis]]
*The [[Suslin's problem|Suslin conjecture]]

The following statements (none of which have been proved false) cannot be proved in ZFC (the Zermelo–Fraenkel set theory plus the axiom of choice) to be independent of ZFC, under the added hypothesis that ZFC is consistent. 

*The existence of [[strongly inaccessible cardinal]]s
*The existence of [[large cardinal]]s
*The non-existence of [[Kurepa tree]]s

The following statements are inconsistent with the axiom of choice, and therefore with ZFC. However they are probably independent of ZF, in a corresponding sense to the above: They cannot be proved in ZF, and few working set theorists expect to find a refutation in ZF. However ZF cannot prove that they are independent of ZF, even with the added hypothesis that ZF is consistent.

*The [[axiom of determinacy]]
*The [[axiom of real determinacy]]
*[[AD+]]