Zermelo Theorem and Axiom of Choice

Grzegorz Bancerek

Warsaw University, Bialystok

Summary.

The article is continuation of [2] and [1], and the goal of it is show that Zermelo theorem (every set has a relation which well orders it - proposition (26)) and axiom of choice (for every non-empty family of non-empty and separate sets there is set which has exactly one common element with arbitrary family member - proposition (27)) are true. It is result of the Tarski's axiom A introduced in [5] and repeated in [6]. Inclusion as a settheoretical binary relation is introduced, the correspondence of well ordering relations to ordinal numbers is shown, and basic properties of equinumerosity are presented. Some facts are based on [4].

MML Identifier: WELLORD2

Contents (PDF format)

Bibliography

[1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.

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

[3] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[4] Kazimierz Kuratowski and Andrzej Mostowski. \em Teoria mnogosci. PTM, Wroc\-law, 1952.

[5] Alfred Tarski. \"Uber Unerreichbare Kardinalzahlen. \em Fundamenta Mathematicae, 30:176--183, 1938.

[6] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[7] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[8] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Journal of Formalized Mathematics, 1, 1989.