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Math Help - The minimal uncountable well-ordered set

  1. #1
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    The minimal uncountable well-ordered set

    Specifically, I'm talking about the set S_{\Omega}, the minimal uncountable well-ordered set, every section of which is countable.

    I'm curious about two things.

    The first is: Are there any (recognizable) representations for this set? It's driving me crazy to come up with an example for the thing.

    The second is: What is the cardinality of this set? It has to be greater than the cardinality of the integers (being that it's uncountable), but I can't see that it can be as big as the cardinality of the real numbers (being that every section of the reals is uncountable) and, by the Continuum Hypothesis, there is no cardinal number between the two.

    Thanks.
    Dan
    Last edited by topsquark; January 28th 2006 at 07:57 AM. Reason: misspelling
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    If you find such a set then it cannot be stricly greater than the cardinality of the integers and strictly less than the cardinality of the continuum, because as you said this would violate the Countinuum hypothesis thus we have two possiblities.

    1)The continuum hypothesis is false and the cardinality of the continuum is NOT the minimal uncountable set.

    2)The continuum hypothesis is true then the minimal cardinality must be the the set of the continuum.

    But as I understand this topic correctly there is NO ANSWER TO YOUR QUESTION
    Because (and I might be wrong on this) the continuum hypothesis is independent from ZFC (even with the Axiom of Choice). Thus, we cannot conclude it is either true or false.

    I think the answer to your question is the continuum.
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    Forum Admin topsquark's Avatar
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    Thank you for the reply ThePerfectHacker. It appears that you are correct.

    It's strange where you find things. I found a couple of theorems about S_{\Omega} in a chapter in my Topology book on connected spaces. Specifically:

    Let L denote the set S_{\Omega} x [0,1) in the dictionary order with its smallest element deleted. (L is an example of what is known as the "long line.")
    1) Theorem: The long line is locally homeomorphic to R.
    2) Theorem: The long line cannot be imbedded in R. Specifically, the long line cannot be imbedded in R^{n} for any n.

    I haven't yet managed to prove either theorem, however I can draw a couple of inferences from them. First, since S_{\Omega} x [0,1) is (essentially) homeomorphic to R then if we call the cardinality of S_{\Omega} a and the cardinality of the reals b, then the homeomorphism predicts ab = b. Using the mathematics of infinite cardinals, the solution for a is a \leq b. The only uncountable infinite cardinal that fits this relation is b. Thus the cardinality of S_{\Omega} is the cardinality of the reals. So ThePerfectHacker is correct. (Not that I doubted it!)

    The other inference I can make is I suspect the long line cannot be imbedded in the reals is due to the order type. (Again, I haven't proven Theorem 2, so I'm only guessing here.) Whatever the case I would imagine the proof of this theorem would allow a construction of the long line space, and thus provide a construction of S_{\Omega}. At least I hope...the chapter IS about connected spaces (in this case local connectivity) and the proof may rely on them, which would mean the proof could well be an existence style proof rather than constructive.

    Anyone have any ideas on how to prove either theorem?

    -Dan

    BTW: I don't know ZFC that well, but I've done a bit of work in axiomatic set theory and I agree. I think I read somewhere that the Continuum Hypothesis is independant from the usual set theory axioms. I know I have read that it is independant from the Axiom of Choice.
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    The concept of cardinal numbers is truly fascinating. I believe some of the future mathematical discoveries will be based on these concepts. Sadly we know almost nothing about them. For example I was reading on Wikipedia (love that site) that there is a Generalized Countinuum Hypothesis that states:
    \aleph_{n+1}=2^{\aleph_n}

    I know this is off topic, but a thing which makes me laugh in math is when mathematicians cannot prove something, but they can prove that it cannot be proven! For example, like the countinuum hypothesis, independece of the parallel postulate from the first four, impossiblity of certain geometric constructions.
    Last edited by ThePerfectHacker; January 31st 2006 at 01:17 PM.
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