cofinality, infinite cardinal

**xianghu21** · March 24th 2010, 09:22 AM

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.

**clic-clac** · March 25th 2010, 05:47 AM

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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