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. --Matt Westwood 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)