# infinite cardinal numbers

• Mar 24th 2010, 09:14 AM
xianghu21
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.
• Mar 25th 2010, 05:34 AM
clic-clac
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.