## 580 -Cardinal arithmetic (2)

February 7, 2009

At the end of last lecture, we showed Theorem 7, König’s lemma, stating that if $\kappa_i<\lambda_i$ for all $i\in I,$ then $\sum_{i\in I}\kappa_i<\prod_i\lambda_i.$ We begin by looking at some  corollaries:

Corollary 8.

1. If $\beta$ is a limit ordinal and $(\kappa_i:i<\beta)$ is a strictly increasing sequence of nonzero cardinals, then $\sum_{\alpha<\beta}\kappa_\alpha<\prod_{\alpha<\beta}\kappa_\alpha.$
2. If $(\kappa_i:i\in I)$ is an $I$-indexed sequence of nonzero cardinals and $\kappa_i<\sum_{j\in I}\kappa_j$ for all $i\in I,$ then $\sum_i\kappa_i<\left(\sum_i\kappa_i\right)^{|I|}.$
3. (Cantor) $\kappa<2^\kappa.$
4. For any infinite $\kappa$, one has $\kappa<\kappa^{{\rm cf}(\kappa)}.$
5. ${\rm cf}(2^\kappa)>\kappa.$