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Constructible universe and large cardinals a la Hugh Woodin

  1. Jan 26, 2012 #1
    Can any math geeks help?

    Refering to the talk,
    (Woodin: plenary talk at the 2010 International Congress of Mathematicians) http://bitcast-a.bitgravity.com/highbrow/livearchive40009/26aug-13.45to14.45.flv
    In Hugh Woodin's 2010 ICM talk on Ultimate L he introduces Godel's constructible universe of sets L then discusses the possibility that the universe of sets V is exactly L and then brings up Scott's theorem which says that if V=L then there are no measurable cardinals.

    Then Woodin says something like, "so V=L is denied by infinity." (About 28 minutes into the video talk linked above.)

    Q. What does he mean by this?

    My problem understanding this statement is that he seems to be saying that large measurable cardinals must somehow be desirable in set theory. Why? Or if I interpret the remark as meaning that any sort of infinite set is denied by V=L then I'm just not groking this, because I always though Godel's L does admit infinite sets, just not these large cardinals.

    So is Woodin really only paraphrasing something like "so V=L is denied by those who would assume large cardinals are useful," or is he stating something stronger or more subtle?

    The best i can make of Woodin's remark is that he means Scott's theorem implies that if V=L then "there are no interesting large cardinals." In which case my question would be what the heck does "an interesting large cardinal" mean?


    PLEASE: no comments necessary from finitists or physicists or anyone else who does not believe in either the consistency or utility of transfinite numbers. I respect your beliefs without needing to agree.
     
  2. jcsd
  3. Jan 26, 2012 #2

    AKG

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    V = L certainly admits infinite sets. In fact, even if V [itex]\neq[/itex] L, V and L have all the same transfinite ordinals. What can happen is that L may not "see" enough of the rest of the universe to be able to tell that certain cardinals have large cardinal properties. There are some large cardinal properties that can be consistent with L, but the larger large cardinals cannot, as per Scott's theorem. Something called [itex]0^{\sharp}[/itex] (read "zero sharp") delineates the boundary between the large cardinals properties which can exist in L and those which can't.

    When Woodin says "V = L is denied by infinity," I think he is passively implying that, yes, the higher infinite (i.e. the large large cardinals) are the natural extension of ZFC; they're not only desirable, their acceptance should be the standard mindset. My impression is that the majority of set theorists would share his view that the large cardinal hierarchy is the way to go, and V=L is not a suitable candidate for a new axiom for set theory.

    That said, although this may be the majority opinion, it's certainly not unanimous, and there may be strong dissenters. I'm not sure who has made a compelling case to reject large cardinals, so I can't suggest any reading in that direction. For impartial-to-favourable takes on the large cardinals, I recommend reading the works of Peter Koellner, Penelope Maddy (she has a series of papers entitled "Believing the Axioms"), and the introductory sections in Kanamori's "The Higher Infinite." I also strongly recommend the following article from New Scientist:

    http://www.newscientist.com/article...e-logic-to-infinity-and-beyond.html?full=true
     
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