Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

## Countable Sets and Hessenberg's Theorem

Grzegorz Bancerek
Warsaw University, Bialystok

### Summary.

The concept of countable sets is introduced and there are shown some facts which deal with finite and countable sets. Besides, the article includes theorems and lemmas on the sum and product of infinite cardinals. The most important of them is Hessenberg's theorem which says that for every infinite cardinal {\bf m} the product ${\bf m} \cdot {\bf m}$ is equal to {\bf m}.

#### MML Identifier: CARD_4

The terminology and notation used in this paper have been introduced in the following articles [15] [10] [18] [17] [2] [19] [8] [7] [12] [3] [5] [4] [16] [1] [6] [11] [9] [13] [14]

Contents (PDF format)

#### Bibliography

[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. Zermelo theorem and axiom of choice. Journal of Formalized Mathematics, 1, 1989.
[6] Grzegorz Bancerek. Cardinal arithmetics. Journal of Formalized Mathematics, 2, 1990.
[7] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[9] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[10] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[11] Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
[12] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[13] Rafal Kwiatek. Factorial and Newton coefficients. Journal of Formalized Mathematics, 2, 1990.
[14] Andrzej Nedzusiak. $\sigma$-fields and probability. Journal of Formalized Mathematics, 1, 1989.
[15] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[16] Andrzej Trybulec. Tuples, projections and Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[17] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[18] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[19] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.