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.

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