# cofinality, infinite cardinal

• Mar 24th 2010, 09:22 AM
xianghu21
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.
• Mar 25th 2010, 05:47 AM
clic-clac
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 $\displaystyle \sum_{\xi\in\text{cf}(\kappa)}\mu_{\xi}=\kappa$