Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

## 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

The terminology and notation used in this paper have been introduced in the following articles [19] [14] [20] [2] [21] [12] [11] [15] [3] [13] [5] [6] [4] [1] [7] [17] [10] [9] [8]

#### 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

#### Bibliography

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[5] Grzegorz Bancerek. The well ordering relations. Journal of Formalized Mathematics, 1, 1989.
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