Axiomatic Set Theory
In 1931, Kurt Gödel proved the first ZFC undecidability result, namely that the consistency of ZFC itself was undecidable in ZFC.
Moreover the following statements are independent of ZFC (shown by Paul Cohen and Kurt Gödel):
- The axiom of constructibility (V = L);
- The generalized continuum hypothesis (GCH);
- The continuum hypothesis (CH);
- The diamond principle (◊);
- Martin's axiom (MA);
- MA + ¬CH.
Note that we have the following chains of implication:
- V = L → ◊
- V = L → GCH → CH.
Assuming that ZFC is consistent, the existence of large cardinal numbers, such as inaccessible cardinals, Mahlo cardinals etc., cannot be proven in ZFC. On the other hand, few working set theorists expect their existence to be disproved.
Read more about this topic: List Of Statements Undecidable In ZFC
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