Axiom of Choice: Finite Character & Maximal Sets

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SUMMARY

The discussion confirms that the axiom of choice is equivalent to a lemma from Teichmüller and Tukey, stating that every family of sets F with finite character possesses a maximal element. A "maximal set" is defined as a set that is not contained in any larger set. This equivalence reinforces the foundational principles of set theory and the implications of the axiom of choice in mathematical contexts.

PREREQUISITES
  • Understanding of set theory concepts, particularly the axiom of choice.
  • Familiarity with finite character in the context of set families.
  • Knowledge of maximal elements in ordered sets.
  • Basic comprehension of mathematical logic and proofs.
NEXT STEPS
  • Research the implications of the axiom of choice in various mathematical fields.
  • Study the concept of finite character in more depth.
  • Explore the relationship between maximal sets and Zorn's Lemma.
  • Investigate the historical context and development of the axiom of choice by Teichmüller and Tukey.
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Mathematicians, logicians, and students of advanced mathematics who are exploring set theory and its foundational axioms.

quasar987
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There is an axiom/lemma from Teichmilles & Tukey that is equivalent to the axiom of choice. It reads,

Every family of sets F that is of finite character (http://en.wikipedia.org/wiki/Finite_character) possesses a maximal element.

I just want to confirm that here, "maximal set" means a set that is itself contained in no greater set?
 
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Yup.
 
Hooray for the axiom of choice!
 

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