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Math Help - cofinality, infinite cardinal

  1. #1
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    cofinality, infinite cardinal

    Show that for each infinite cardinal \kappa, \kappa <_c \kappa^{\text{cf}(\kappa)}.

    Notation: \text{cf} denotes the cofinality. I know some properties of \text{cf}(\kappa). They may be helpful.

    \text{cf}(\kappa) \leq_c \kappa
    For each infinite cardinal number \kappa, \text{cf}(2^{\kappa}) >_c \kappa.

    However, I do not see how to prove this. Any hints would be great. Thanks.
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  2. #2
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    Here one of Koenig's lemmas helps:

    Consider (\mu_i),\ (\lambda_i) two families of cardinals indexed by I, such that for any i\in I,\ \mu_i<\lambda_i, then:

    \sum_I\mu_i<\prod_I\lambda_i

    Use this result with the fact that given a cardinal \kappa:
    \text{cf}(\kappa) is the lowest cardinal such that there exists a family (\mu_{\xi})_{\xi\in\text{cf}(\kappa)} with for all \xi\in\text{cf}(\kappa),\ \mu_{\xi}<\kappa and \sum_{\xi\in\text{cf}(\kappa)}\mu_{\xi}=\kappa
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