Journal of Formalized Mathematics
Volume 1, 1989
University of Bialystok
Copyright (c) 1989 Association of Mizar Users

## Sequences of Ordinal Numbers

Grzegorz Bancerek
Warsaw University, Bialystok

### Summary.

In the first part of the article we introduce the following operations: On $X$ that yields the set of all ordinals which belong to the set $X$, Lim $X$ that yields the set of all limit ordinals which belong to $X$, and inf $X$ and sup $X$ that yield the minimal ordinal belonging to $X$ and the minimal ordinal greater than all ordinals belonging to $X$, respectively. The second part of the article starts with schemes that can be used to justify the correctness of definitions based on the transfinite induction (see [1] or [4]). The schemes are used to define addition, product and power of ordinal numbers. The operations of limes inferior and limes superior of sequences of ordinals are defined and the concepts of limit of ordinal sequence and increasing and continuous sequence are introduced.

#### MML Identifier: ORDINAL2

The terminology and notation used in this paper have been introduced in the following articles [6] [3] [7] [8] [2] [1] [5]

Contents (PDF format)

#### Bibliography

[1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[4] Kazimierz Kuratowski and Andrzej Mostowski. \em Teoria mnogosci. PTM, Wroc\-law, 1952.
[5] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[6] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[7] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[8] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.