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Continuum Hypothesis is true?

  1. Aug 2, 2011 #1

    mathman

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  3. Aug 2, 2011 #2

    micromass

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    Woodin is a very good set theorist, so he probably did something quite interesting. But the article doesn't really tell me what it is that Woodin did. I get the impression that he built another constructible universe [itex]\mathbb{L}[/itex] which seems to encompass a lot of current mathematics. This wouldn't solve the continuum hypothesis of course, the continuum hypothesis has been proved unsolvable.

    I'm really interested in reading a more advanced article on the matter, to see what it's all about.
     
  4. Aug 2, 2011 #3
    Thanks for the great article, mathman! Personally, I've always found the undecidability of certain statements to be an unsatisfying answer, so maybe this idea can change that.
     
  5. Aug 2, 2011 #4
    In ZFC set theory.

    My personal reaction - it seems like you could create a set theory universe in which the continuum hypothesis is true, false, or undecideable. But does this say anything really over the truth of the continuum hypothesis itself?

    Perhaps a larger question - besides the issue of consistency, how do you know which set theory is "right"?
     
  6. Aug 2, 2011 #5
    Now you're asking if math is Platonic. And the last time somebody asked that, the thread got carted off to the Philosophy section.

    https://www.physicsforums.com/showthread.php?t=514581

    By the way, if anyone's unfamiliar with Freiling's axiom of symmetry, it's an easily understandable and intuitively plausible axiom that makes CH false.

    http://en.wikipedia.org/wiki/Freiling's_axiom_of_symmetry

    The statement of the axiom is easy to understand; as is the proof that the axiom implies the negation of CH. It's an interesting example, and far more understandable than Woodin's work is ever going to be to most of us (speaking for myself here.)


    CH will always be provable in some axiom systems and its negation provable in others. The goal is to find an intuitively appealing set of axioms that settles the issue. I would be quite surprised if Woodin's framework is intuitively appealing to anyone outside of specialists in set theory. Here is an article about Woodin's Ultimate L. It's very technical and presumes a background in advanced set theory.

    http://caicedoteaching.wordpress.com/2010/10/19/luminy-hugh-woodin-ultimate-l-i/

    Wikipedia has nothing on Ultimate L yet ... now that's an article I'd like to read!
     
    Last edited: Aug 2, 2011
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