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Math Help - How many degrees of infinity?

  1. #1
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    How many degrees of infinity?

    Hi,

    I know that there are "infinitely many" degrees of infinity. The steps for the proof I have seen:

    1. For any cardinal, k, k<2^k
    2. So we have countably many different degrees of infinity by considering; (aleph_null)<2^(aleph_null)<2^(2^(aleph_null))<...

    I was wondering if it is know how many degrees of infinity there are (eg are there only countably many, uncountably many, even more?)

    MY THOUGHTS:

    (Assuming AC) The cardinals are identified with the ordinals so this is the same question as asking how many ordinals there are. But the ordinals form a proper class (a class which is not a set) so can we even find a cardinality for the ordinals?


    Thanks in advance for any replies.
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  2. #2
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    Re: How many degrees of infinity?

    Quote Originally Posted by oOoOo View Post
    (Assuming AC) The cardinals are identified with the ordinals so this is the same question as asking how many ordinals there are.
    No, each cardinal is an ordinal, but not every ordinal is a cardinal.

    However, for each ordinal k, there is the cardinal aleph_k. So in that sense, if there are uncountably many ordinals then there are uncountably many cardinals. And there are uncountably many ordinals in the sense that at least there is a set of ordinals that is uncountable. So there are uncountably many cardinals in the sense that at least there is a set of cardinals that is uncountable.

    Quote Originally Posted by oOoOo View Post
    (But the ordinals form a proper class (a class which is not a set) so can we even find a cardinality for the ordinals?
    Right, proper classes do not have a cardinality.

    There is no cardinality of the cardinals, but there are uncountably many cardinals.
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  3. #3
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    Re: How many degrees of infinity?

    Thanks for the reply.

    Yes, I should have said, the initial ordinals (ordinals not equnumerous with any smaller ordinal) are identified with the cardinals? Is that correct?

    The cardinals form a proper class so we cannot assign a cardinality to them. As you said, the cardinals are in some sense bigger than aleph_null, can we say similar for any given cardinality?
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    Re: How many degrees of infinity?

    Quote Originally Posted by oOoOo View Post
    the initial ordinals (ordinals not equnumerous with any smaller ordinal) are identified with the cardinals? Is that correct?
    Yes.

    Quote Originally Posted by oOoOo View Post
    The cardinals form a proper class so we cannot assign a cardinality to them. As you said, the cardinals are in some sense bigger than aleph_null, can we say similar for any given cardinality?
    Of course, since, as you said, the class of cardinals is a proper class.

    EDIT:

    Oops, the below is good, which I just posted, but it answers a different question.

    For any cardinal k greater than 0, there's no cardinal j such that there are only j number of sets of cardinality k.

    Suppose you have j number of sets of cardinality k where k is greater than 0. Take some cardinal p greater than j. Take just ONE of the sets of cardinality k. For some member x in that set, suppose it's not a cardinal number itself (if x is a cardinal, then swap in some non-cardinal not already in the set). Now form p number of different sets of cardinality k by swapping in a different member of p.

    Indeed, for any cardinal k greater than 0, the class of sets having cardinality k is a proper class.
    Last edited by MoeBlee; July 18th 2011 at 02:07 PM.
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  5. #5
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    Re: How many degrees of infinity?

    One could take the original question in this thread onto another track in asking oOoOo what axioms he/she would like to assume. Obviously one is assuming ZFC, including the axiom of infinity and the power set axiom, but if one is assuming some additional axioms of the type that are often (at least implicitly) assumed in doing a lot of mathematics (the existence of an inaccessible, or of an uncountable measurable, etc), some large cardinal axioms or some negations of some large cardinal axioms, then the answer as to how many there are will depend on which combination you choose. Being uncountable is only the start of the fun.
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  6. #6
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    Re: How many degrees of infinity?

    The axioms I would like to assume are ZFC (this is all I have met in set theory so far).

    Could anyone verify/point out an error in this argument:

    Let k be a cardinal (i.e. an initial ordinal)

    Let alpha be an initial ordinal which is strictly greater than k (we can prove that this always exists)

    Then the set A={aleph_beta : beta is an element of alpha} is a set of cardinals which is equinumerous with alpha.

    Hence for any cardinal k, there is a set of cardinals, A, with cardinality strictly greater than k.

    So, in some sense, the class of cardinals is larger than any degree of infinity?
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    Re: How many degrees of infinity?

    oOoOo: Your last line
    the class of cardinals is larger than any degree of infinity
    does not follow from your penultimate line
    for any cardinal k, there is a set of cardinals, A, with cardinality strictly greater than k
    , even though both are valid. Of course, the last line is valid if we take liberties, and say that "A is larger than B" means "B is a member of A", and that a "degree of infinity" means a cardinal. Then the conclusion is a tautology. (If you used "larger than" as a partial order, as a relation among sets, then you can't have a proper class involved in a statement with this phrase. But I am being liberal with your "in some sense".) If "degree of infinity" means ordinal, it is not much to come to the same conclusion.
    The argument up to the penultimate line is OK.
    But to jump from the penultimate line to the last line, you would need to assume that A was the class of all cardinals (and take liberties with the set notation), which would mean that alpha was also the (proper) class of all cardinals, which would mean that it could not be an ordinal. So this step doesn't work.
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    Re: How many degrees of infinity?

    On second thought, there is something we should prove here:

    If S is a set of cardinals, then there is a set of cardinals S' such that card(S') > card(S).

    And oOoOo did that with his argument.
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    Re: How many degrees of infinity?

    Quote Originally Posted by oOoOo View Post
    Let k be a cardinal (i.e. an initial ordinal)

    Let alpha be an initial ordinal which is strictly greater than k (we can prove that this always exists)

    Then the set A={aleph_beta : beta is an element of alpha} is a set of cardinals which is equinumerous with alpha.

    Hence for any cardinal k, there is a set of cardinals, A, with cardinality strictly greater than k.
    Looks good to me.
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    Re: How many degrees of infinity?

    delete
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    Re: How many degrees of infinity?

    Quote Originally Posted by oOoOo View Post
    So, in some sense, the class of cardinals is larger than any degree of infinity?
    I'd leave that out. It's not rigorous what is meant by a class being larger than any degree of infinity.

    The important theorem is the one you proved: For any set S of cardinals, there is a set S' of cardinals such that card(S') > card(S).
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    Re: How many degrees of infinity?

    Quote Originally Posted by nomadreid View Post
    If "degree of infinity" means ordinal, it is not much to come to the same conclusion.
    So what exactly does "degree of infinity" mean? I am just taking it to mean an infinite cardinal.
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    Re: How many degrees of infinity?

    Thanks for all your replies.

    I am still struggling to see what exactly is the answer to my original question. Is the answer simply that we cannot assign a size (cardinality) to the collection of cardinals because they form a proper class?

    This seems unsatisfactory to me and is leading me to want to say something like: "so the size of the collection of degrees of infinity (infinite cardinals) is absolutely infinite".

    I realise this is not very mathematical! But it feels unsatisfactory to have built up this theory of the transfinite only to be discovering new things that we still cannot assign a size to.
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    Re: How many degrees of infinity?

    Quote Originally Posted by oOoOo View Post
    we cannot assign a size (cardinality) to the collection of cardinals because they form a proper class?
    Yes, that's clear, right?

    Quote Originally Posted by oOoOo View Post
    it feels unsatisfactory to have built up this theory of the transfinite only to be discovering new things that we still cannot assign a size to.
    In Z set theories, there IS no thing such that every cardinal is a member of it. In Z set theories, there is no issue of assigning a cardinality to such a thing that has all the cardinals as members, since there is no such thing. In class theories such as NBG, there is the proper class of all and only the cardinals, but there does not exist a bijection between that class and any cardinal, so there is no object to assign as the cardinality of the class of cardinals.

    Is that situation any more baffling than, for example, in the reals, for nonzero x there is no number y such that 0*y = x?
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  15. #15
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    Re: How many degrees of infinity?

    So what exactly does "degree of infinity" mean? I am just taking it to mean an infinite cardinal.
    oOoOo: Well, it was your term. It is a rather loose term as mathematical terms go, but as it is often taken, as you are doing, to be the cardinality, I answered in that vein.

    it feels unsatisfactory to have built up this theory of the transfinite only to be discovering new things that we still cannot assign a size to.
    MoeBlee's answer was good; I might add that if you (oOoOo) feel the need to have something that you can assign a size to somehow, perhaps you would like to take a universe V_\kappa for a cardinal \kappa which is a "slice" of the universe V; V_\kappa is a class in one context but, in the context of a larger universe, will be a set and can then be assigned a size in that larger universe. Such "slices" tend to be sufficient for most purposes. See for starters
    Von Neumann universe - Wikipedia, the free encyclopedia,
    Large cardinal - Wikipedia, the free encyclopedia
    List of large cardinal properties - Wikipedia, the free encyclopedia
    Last edited by nomadreid; July 19th 2011 at 09:20 PM.
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