>>11512989>Redpill me on continuum hypothesis. I know it's independent of ZFC but surely we know more about it now. I heard Woodin has interesting things to say about it?
The continuum hypothesis (CH) is the assertion that any infinite set which is too big to be matched one-to-one with the integers has at least as many elements as the set of all real numbers.
The generalized continuum hypothesis (GCH) asserts that the smallest infinite set which is too big to be matched one-to-one to an infinite set S can be matched one-to-one with the set of all subsets of S.
The continuum hypothesis was formulated by Georg Cantor as a conjecture in his theory of ordinal and cardinal infinities. David Hilbert listed it as the first of his 23 problems for 20th century mathematics.
In 1940 Kurt Godel, extending earlier ideas of Hilbert, proved that it is consistent with standard set theory that GCH is true.
In 1963, Paul Cohen proved that it is also consistent with standard set theory to assume that CH is false. Cohen's method made it clear that CH and GCH are also unprovable in any extension of set theory which only adds axioms which assert the consistency of previous axioms.
Because the CH is independent of standard axiom systems, most mathematicians consider it an undecidable proposition with no definite truth value. Probabilistic intuition leads many to also view it as more false than true. Mathematicians with a more platonist view consider it an open question, to be settled by new axioms.