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Thread: infinite cardinal numbers

  1. #1
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    infinite cardinal numbers

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

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

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

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

    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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    Hi

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

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

    Conclude by showing the reversed inequality.
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