Results 1 to 7 of 7

Math Help - Infinite ordinal exponentiation doesn't raise cardinality

  1. #1
    Newbie
    Joined
    Oct 2009
    From
    New York, NY
    Posts
    6

    Infinite ordinal exponentiation doesn't raise cardinality

    I've been having trouble with the following:

    Assume that a, b are ordinals with 1< min(a, b) and omega <= max(a, b). Then |a^b| =max(a, b).

    Anything would help, thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Jan 2008
    Posts
    19
    Quote Originally Posted by culturalreference View Post
    I've been having trouble with the following:

    Assume that a, b are ordinals with 1< min(a, b) and omega <= max(a, b). Then |a^b| =max(a, b).

    Anything would help, thanks.
    This isn't true as phrased. You don't have to look far for a counter example. a=2 b=anything, |a^b| will always be strictly larger than a and b. Did you forget some hypothesizes?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2009
    From
    New York, NY
    Posts
    6
    Quote Originally Posted by wgunther View Post
    This isn't true as phrased. You don't have to look far for a counter example. a=2 b=anything, |a^b| will always be strictly larger than a and b. Did you forget some hypothesizes?
    your counter example is not true if b = anything greater than or equal to omega, like it has to be, by the hypothesis.

    The hypothesis says that one or both of the ordinals is greater than or equal to omega, and if one is not greater than omega then it can't be less than two.

    thanks.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jan 2008
    Posts
    19
    Quote Originally Posted by culturalreference View Post
    your counter example is not true if b = anything greater than or equal to omega, like it has to be, by the hypothesis.

    The hypothesis says that one or both of the ordinals is greater than or equal to omega, and if one is not greater than omega then it can't be less than two.

    thanks.
    2^\kappa=|\wp(\kappa)|>\kappa for all kappa, even omega. 2^omega has cardinality continuum which can be very large, def larger than omega. The thing that is similar to what you are saying has extra hypothesizes about cofinality. So I'm still not sure what your question means.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    @wgunther: 2^\omega=\omega this is ordinal exponentiation, not cardinal exponentiation.

    @culturalreference: Have you tried by ordinal induction?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jan 2008
    Posts
    19
    Quote Originally Posted by clic-clac View Post
    @wgunther: 2^\omega=\omega this is ordinal exponentiation, not cardinal exponentiation.

    @culturalreference: Have you tried by ordinal induction?
    My fault! Must have missed a word. Apologies
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Oct 2009
    From
    New York, NY
    Posts
    6
    clic-clac: thanks for the hint, but i'm struggling with what the mapping would go to. In other words,

    so i show by induction that a^n = a (for a > omega), for n < b < omega. Then i feel like the next step has to be to show a map from U a^n ---> something, but i feel like this map can't just be to omega. does it have to be to omega x omega, or a x omega, or...?

    Thanks, and sorry if this is muddled.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: November 7th 2010, 05:06 PM
  2. derivative of x raise to log
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 27th 2010, 11:49 AM
  3. Ordinal exponentiation
    Posted in the Discrete Math Forum
    Replies: 11
    Last Post: September 24th 2010, 08:32 AM
  4. Cardinality if infinite sets
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: May 24th 2010, 12:43 PM

Search Tags


/mathhelpforum @mathhelpforum