On Powers of Cardinals

Grzegorz Bancerek

IM PAN, Warsaw, Warsaw University, Bialystok

Summary.

In the first section the results of [18, axiom (30)]\footnote {Axiom (30)\quad -\quad $n = \{k\in{\Bbb N}: k < n\}$ for every natural number $n$.}, i.e. the correspondence between natural and ordinal (cardinal) numbers are shown. The next section is concerned with the concepts of infinity and cofinality (see [8]), and introduces alephs as infinite cardinal numbers. The arithmetics of alephs, i.e. some facts about addition and multiplication, is present in the third section. The concepts of regular and irregular alephs are introduced in the fourth section, and the fact that $\aleph_0$ and every non-limit cardinal number are regular is proved there. Finally, for every alephs $\alpha$ and $\beta$ $$\alpha^\beta = \left\{ \begin{array}{ll} 2^\beta,& {\rm if}\ \alpha\leq\beta,\\ \sum_{\gamma<\alpha}\gamma^\beta,& {\rm if}\ \beta < {\rm cf}\alpha\ {\rm and} \ \alpha\ {\rm is\ limit\ cardinal},\\ \left(\sum_{\gamma<\alpha}\gamma^\beta\right)^{\rm cf\alpha},& {\rm if\ cf}\alpha \leq \beta \leq \alpha.\\ \end{array}\right.$$ \\ Some proofs are based on [16].

MML Identifier: CARD_5

Contents (PDF format)

1. Results of \cite[axiom (30)]{AXIOMS.ABS}
2. Infinity, alephs and cofinality
3. Arithmetics of alephs
4. Regular alephs
5. Infinite powers

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