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Thread: cofinality, infinite cardinal

  1. #1
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    cofinality, infinite cardinal

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

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

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

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

    $\displaystyle \sum_I\mu_i<\prod_I\lambda_i$

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