# Equivalents of the axiom of choice

```@inproceedings{Grabczewski2001EquivalentsOT,
title={Equivalents of the axiom of choice},
author={Krzysztof Grabczewski},
year={2001}
}```
Sets, Models and Proofs
• Mathematics
• 2018
This textbook provides a concise and self-contained introduction to mathematical logic, with a focus on the fundamental topics in first-order logic and model theory, and explores the formal axiom system of Zermelo and Fraenkel before concluding with an extensive list of suggestions for further study.
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 4 (17) 2001
Logic as a formal mathematical theory is interesting in itself. It poses problems, like any other body of knowledge. Some of them, especially questions concerning axiomatization, consistency,
The Axiom of Choice, Well-Ordering, and Well-Classification
• H. H. Giv
• Mathematics, Economics
Am. Math. Mon.
• 2015
A simple characterization of those choice functions on P*(X) that are induced by a well-ordering on X is given and this is finally proved to be an equivalent form of the axiom of choice.
TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY
• Z. Weber
• Philosophy
The Review of Symbolic Logic
• 2010
This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic, and indicates how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.
• Philosophy
• 2007
We give some lectures on the work on formal logic of Jacques He rbrand, and sketch his life and his influence on automated theorem proving. The intended audience ranges from students interested in
Łoś's theorem and the axiom of choice
In set theory without the Axiom of Choice ( AC ), we investigate the problem of the placement of Łoś's Theorem ( LT ) in the hierarchy of weak choice principles, and answer several open questions
The Bulletin of Symbolic Logic
• Philosophy
• 2014
We try to recast in modern terms a choice principle conceived by Beppo Levi, who called it the Approximation Principle (AP). Up to now, there was almost no discussion about Levi’s contribution, due
Mechanizing set theory
• Mathematics
Journal of Automated Reasoning
• 2004
Fairly deep results of Zermelo-Frænkel set theory have been mechanized using the proof assistant Isabelle, and the equivalence of 7 formulations of the Well-ordering Theorem and 20 formulations of AC is proved.
The cardinal inequality α2 < 2α
• Mathematics
• 2011
Abstract In ZFC set theory (i.e., Zermelo-Fraenkel set theory with the Axiom of Choice (AC)) any two cardinal numbers are comparable. However, this may not be valid in ZF (i.e., Zermelo-Fraenkel set

## References

SHOWING 1-3 OF 3 REFERENCES
Equivalents of the Axiom of Choice II
• Economics
• 1985
Set Forms. The Well-Ordering Theorem. The Axiom of Choice. The Law of the Trichotomy. Maximal Principles. Forms Equivalent to the Axiom of Choice Under the Axioms of Extensionality and Foundation.
Mechanizing Set Theory: Cardinal Arithmetic and the Axiom of Choice
• Mathematics
ArXiv
• 1996
Fairly deep results of Zermelo-Frenkel set theory have been mechanized using the proof assistant Isabelle, and the equivalence of 7 formulations of the Well-ordering Theorem and 20 formulations of AC is proved.