Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

**Grzegorz Bancerek**- IM PAN, Warsaw, Warsaw University, Bialystok

- 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].

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

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Grzegorz Bancerek.
Sequences of ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Grzegorz Bancerek.
The well ordering relations.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Grzegorz Bancerek.
Zermelo theorem and axiom of choice.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Grzegorz Bancerek.
Cardinal arithmetics.
*Journal of Formalized Mathematics*, 2, 1990. - [8]
Grzegorz Bancerek.
Consequences of the reflection theorem.
*Journal of Formalized Mathematics*, 2, 1990. - [9]
Grzegorz Bancerek.
Increasing and continuous ordinal sequences.
*Journal of Formalized Mathematics*, 2, 1990. - [10]
Grzegorz Bancerek.
K\"onig's theorem.
*Journal of Formalized Mathematics*, 2, 1990. - [11]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [16] Wojciech Guzicki and Pawel Zbierski. \em Podstawy teorii mnogosci. PWN, War\-sza\-wa, 1978.
- [17]
Andrzej Nedzusiak.
$\sigma$-fields and probability.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Andrzej Trybulec.
Strong arithmetic of real numbers.
*Journal of Formalized Mathematics*, Addenda, 1989. - [19]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [20]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

[ Download a postscript version, MML identifier index, Mizar home page]