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Math Help - Set theory question.

  1. #1
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    Set theory question.

    Hi all,

    In Kunen's book, one of the exercises is the following:

    Show that for \kappa > \omega, |H( \kappa)| = 2^{< \kappa}.

    But, H( \omega) = V( \omega) and |V( \omega)| = \omega. Also, 2^{< \omega}= \omega (because 2^{< \omega} is the cardinality of the union of omega many finite sets (the characteristic functions for each n)).

    Is this wrong?

    Thanks in advance!

    Sam
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  2. #2
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    The notation used in this post is so non-standard, I think that you need to define what each symbol means or how it is used.
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  3. #3
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    Ok, so:

    \kappa is used as a variable over cardinals.
    | x| denotes the cardinality of x.
    \alpha^{<\beta} is the cardinality of the union of \alpha^{\delta} for \delta < \beta.
    H( \kappa) denotes the set of sets which are hereditarily of cardinality < \kappa.
    V( \alpha) is the \alpha^{th} level in the cumulative hierarchy.

    Hope this helps.

    Please say if I've left anything important out.

    Regards
    Sam
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  4. #4
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    Just a tangential point: the notation used is not non-standard as far as I am aware. Kunen's book is a standard text, which contains all of this notation except V(\alpha). In Kunen R is used instead of V. In most modern treatments, though, V is used. See Von Neumann universe - Wikipedia, the free encyclopedia, for an example.

    Regards
    Sam
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