Talk:Continuum Hypothesis

I decided to put this in open questions because I'm not convinced if proving it to be unprovable really counts as a solution to the problem. If you disagree, feel free to change it, but please state your reasons here. --Cynic (talk) 04:36, 20 November 2009 (UTC)

I'm not sure ... the question has been settled, it's just that the answer wasn't as "simple" as stated in the question. Similar with the AoC. Unfortunately I don't know enough yet to understand it enough to make a call. We'll leave it here for the moment and see what it looks like when the brain is bigger. --Prime.mover 06:24, 20 November 2009 (UTC)

I decided I was being dumb and removed it from the open questions category. --Cynic (talk) 03:07, 28 December 2009 (UTC)

The above isn't dumb at all. Whether it's settled or not depends on what exactly the question is. If the question is "Does CH follow from ZFC?" then it is resolved: it does not follow, and furthermore neither does $\neg$CH.

But, Cantor conjectured CH decades before axiomatic set theory and ZFC came into existence. It appeared on Hilbert's problem list before ZFC existed as well. So it's pretty reasonable to argue that the "ZFC form" of the question isn't necessarily what Cantor had in mind.

This can go touch a bit on math philosophy, but the other view on the question is that it must have a "real answer": either these sorts of sets exist or they do not. With this view, all the independence results have shown is that ZFC doesn't really "capture" set theory as Cantor imagined it. The possibility remains that another "better" axiomatization of set theory (besides just ZFC+CH) could be proposed in which CH can be decided. It would be up to historians to decide whether this system would really capture what Cantor had in mind to see if it answered the question he was asking.

-- Qedetc 16:02, 12 June 2011 (CDT)

That coincides with my philosophical view of this. I've taken the liberty of adding it back into "Open Questions" category. --prime mover 16:32, 12 June 2011 (CDT)

Has there been any progress on including the proofs of the unresolvable nature of the CH in ZFC? If not, does anyone know where I can find a summary of the argument that might be accessible to someone with an undergraduate degree in mathematics? If that's impossible, could anyone point me towards resources that would inform me enough that I could approach a summary of the aforementioned proofs? I'm fascinated by the CH and by the implications of unresolvable assertions within ZFC, but I haven't been able to find any satisfactory resources. --aleph_one 15:28, 3 April 2014

Cohen's proof of independence relies on the extremely difficult technique of "forcing" to construct a set-theoretic universe violating $2^{\aleph_0} = \aleph_1$. References for this can be found using that term. A quite accessible though labour-intensive (lots of stuff left to the reader, tough exercises) approach to the subject is to work through T. Jech's Set Theory, Chapter 14 of which presents Cohen's proof. Hope that helps.

(As (the house-style conforming part of) $\mathsf{Pr} \infty \mathsf{fWiki}$ barely manages to veer into the direction of logic beyond the basics of Propositional Logic, it will be quite some time before nontrivial set theory will be on the cards.) — Lord_Farin (talk) 20:33, 3 April 2014 (UTC)

Axiom namespace?

Should probably move to Axiom:Continuum Hypothesis, as its status is comparable to other axioms. --barto (talk) (contribs) 14:04, 6 November 2017 (EST)