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"Pursuing that challenge, Kurt Gödel showed that the known axioms of mathematics are insufficient to disprove the hypothesis." Is this related to Godel's Incom
by ucarion 13y ago
"Pursuing that challenge, Kurt Gödel showed that the known axioms of mathematics are insufficient to disprove the hypothesis."
Is this related to Godel's Incompleteness Theorem? Are there any non-mathematician's explanations as to how Godel proved this?
- ultrafilter 13y agoThere's no relation to his incompleteness theorems. Gödel proved that every universe V of sets that satisfies the (ZFC) axioms has a subuniverse L that satisfies both the axioms and the continuum hypothesis (CH). L is constructed by transfinite recursion. At each stage of the construction, the only new sets you admit are the definable subsets of the set of all sets you admitted in earlier stages. http://en.wikipedia.org/wiki/Godel%27s_constructible_universe http://en.wikipedia.org/wiki/Godel%27s_constructible_univers... Unfortunately, proving that L satisfies ZFC and CH is a very delicate and technical matter. On the other hand, it can be fun because it's extremely meta. One of the steps of the argument is proving that L is the L of L. I learned the proof from a graduate textbook chapter titled "defining definability."