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Math Help - ORD subset of L; |ORD| uncountable; |L| countable --?

  1. #1
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    ORD subset of L; |ORD| uncountable; |L| countable --?

    I have one or more of the following steps incorrect. Which one(s)?
    (a) Gödel's constructible universe L is the minimal model for ZFC, hence countable.
    (b) The set of ordinals ORD is uncountable.
    (c) ORD is a subset of L, hence |ORD| is less than or equal to |L|.
    The contradiction is evident. ????
    Thanks.
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    Quote Originally Posted by nomadreid View Post
    I have one or more of the following steps incorrect. Which one(s)?
    (a) Gödel's constructible universe L is the minimal model for ZFC, hence countable.
    (b) The set of ordinals ORD is uncountable.
    (c) ORD is a subset of L, hence |ORD| is less than or equal to |L|.
    The contradiction is evident. ????
    Thanks.

    There is no "the set of ordinals" in ZFC anymore than there is a "set of all sets"...take a peek at
    Constructible universe - Wikipedia, the free encyclopedia

    Tonio
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    Thanks, Tonio. Now that I have more sleep than when I posted this, I realize that all three points were incorrect. As you say, ORD is not a set, and L is not the minimal model, and ORD is a subclass of L, and talking of cardinalities of classes is nonsense. Oops.
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    Maybe Skolem paradox is relevant.
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    Thanks, emakarov. The Skolem paradox is one of those things that makes physicists think that mathematicians are crazy. My large oversight was that the minimal, hence countable model was not L but rather L_k for some ordinal k, and then there is a countable (in V) set of ordinals W= L_k intersect ORD such that W is a subset of L_k. I was mixing viewpoints: from inside and outside of L_k. This is of course the main reason Skolem's "paradox" seems paradoxical.
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