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Math Help - infinite cardinal numbers

  1. #1
    Newbie
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    Nov 2008
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    18

    infinite cardinal numbers

    For every family of infinite cardinal numbers (i \mapsto \kappa_i) on a non-empty index set I,

    \sum_{i \in I} \kappa_i =_c \text{max}(|I|, \text{sup}(\{ \kappa_i | i \in I \})).

    I do not see how to prove this. We have proved a theorem that says that:

    For every indexed family of sets (i \rightarrow \kappa_i)_{i \in I} and ever infinite \kappa, if |I| \leq_c \kappa and for each \kappa_i \leq_c \kappa, then \sum_{i \in I} \kappa_i \leq_c \kappa.

    I am not sure if this theorem helps or not though. I need help on this one. Thanks.
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  2. #2
    Senior Member
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    Nov 2008
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    Hi

    Not only the theorem you stated does help, but it almost completely proves your assertion

    Define \kappa := \max\{I,\sup\{\kappa_i\ ;\ i\in I\}\}. You can easily check that the theorem's conditions are fulfilled, hence \sum_{i\in I}\kappa_i\leq\kappa.

    Conclude by showing the reversed inequality.
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