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, granting that ZF is consistent:
- The axiom of choice
- The continuum hypothesis and the generalised continuum hypothesis
- The Suslin conjecture
The following statements (none of which have been proved false) cannot be proved in ZFC to be independent of ZFC, even if the added hypothesis is granted that ZFC is consistent. However, they cannot be proved in ZFC (granting that ZFC is consistent), and few working set theorists expect to find a refutation of them in ZFC.
- The existence of strongly inaccessible cardinals
- The existence of large cardinals
- The non-existence of Kurepa trees
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+
Read more about this topic: Independence (mathematical Logic)
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