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Philosopher Chris Gifford (Bristol) writes:
I have been hearing some interesting news about developments in the philosophy of mathematics. Can you do a thread on your blog on the question: "Has the Continuum Hypothesis been Solved? If, so, who solved it? If not, what has been the best attempt?".
Informed readers, please educate us!
The linked article has a pretty good summary of the history and status of CH. What more does Dr. Gifford want?
BL COMMENT: I should note I added the link, so perhaps it does answer the question, but I would be glad to hear from others.
It would have been nice to know what Gifford was referring to by the "interesting news". Certainly it's not anything that has been discussed on the Foundations of Math mailing list (http://www.cs.nyu.edu/mailman/listinfo/fom) in the last few years.
It would also be nice to know what he means by "solved". Yes, Woodin has introduced axioms which imply that CH is false, but this is no different from Godel's V=L axiom which implies it is true. But there is not any consensus among people working in foundations of math whether V=L or Woodin's axioms should be added to ZFC.
There was an article in Scientific American a few years ago on Woodin's work which used terms like "dispute" and "settled," perhaps that is what Gifford meant.
As Sara points, people keep coming up with new, natural, sometimes pretty axioms which imply either CH or its negation, but I don't think this rises to the level of "news" that would interest the general public. (In the 80s people were talking about a simple axiom, the Axiom of Symmetry, which by an old result of Sierpinski is equivalent to not-CH, and which could apparently let you predict the future [or maybe not-AS let you predict the future], but that was a bit of a stretch.)
Woodin'swork is intimidatingly technical, but to my knowledge is the most likely "news" of the past couple of decades that MIGHT be taken as implying a resolution of the Continuum Problem. For a (more) accessible account of Woodin's basic ideas, I'd recommend Peter Koellner's article in "Philosophia Mathematica," v.14 (2006), pp. 153-188. This, and a number of other relevant things (like: Stanford Encyclopedia of Philosophy articles by Koellner) seems to be downloadable from Koellner's website,
http://logic.harvard.edu/koellner/
…
Non-bibliographical comment: Given the Gödel-Cohen independence result, it is clear that any resolution will have to depend on making plausible some axiom that goes beyond the standard ZFC axioms. What sort of argument could make such an axiom seem plausible is unclear, though the "reflection" arguments that motivate many strong axioms of infinity (though not, alas, it seems, ones which decide CH) seem to convince many set theorists: it is, in other words, far frombeinga priori obvious that there CAN'T be a convincing argument for new axioms. I think it is fair to say that Woodin's technical work suggests that , in some sense, ~CH "coheres" better with our intuitive idea of the set-theoretic universe than CH does: it's part of a tidier, more orderly, overall picture.
Thank you for your comments.
Woodin's current work, which has been developing over the last ~6 years, implies that CH is true.
His earlier work on forcing axioms and generic absoluteness, which the above commenters are discussing, turned out to have technical implications which he believes indicate its falsity.
Thank you, DB.
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