Axiom of Choice: Finite Character & Maximal Sets

Thread starter quasar987
Start date Dec 9, 2007
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Axiom Choice

In summary, the Axiom of Choice is a mathematical principle that allows for the selection of an element from each set in an infinite collection of sets. It is used in set theory and has applications in various areas of mathematics. The term "finite character" refers to the property of a set containing a finite number of elements. Maximal sets cannot be extended by adding more elements and are important in proving the existence of choice functions. The Axiom of Choice is one of the axioms of the Zermelo-Fraenkel set theory and is used to prove the existence of certain mathematical objects. Some criticisms of the Axiom of Choice include counterintuitive results and the argument that it is not necessary for most mathematical proofs.

Dec 9, 2007

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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Dec 10, 2007

morphism

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Yup.

Dec 10, 2007

Coin

Hooray for the axiom of choice!

1. What is the Axiom of Choice?

The Axiom of Choice is a mathematical principle that allows for the selection of an element from each set in an infinite collection of sets, even if there is no rule or pattern for making the selection. It is an important tool in set theory and has applications in various areas of mathematics.

2. What is meant by "finite character" in the Axiom of Choice?

The term "finite character" refers to the property of a set that it only contains a finite number of elements. In the context of the Axiom of Choice, this means that the sets involved in the axiom are finite, as opposed to infinite.

3. What are maximal sets in relation to the Axiom of Choice?

In the Axiom of Choice, a maximal set is a set that cannot be extended by adding any more elements to it. This means that every proper subset of a maximal set is strictly smaller than the set itself. Maximal sets are important in proving the existence of choice functions.

4. How does the Axiom of Choice relate to Zermelo-Fraenkel set theory?

The Axiom of Choice is one of the axioms of the Zermelo-Fraenkel set theory, which is a widely accepted foundation for mathematics. It is used to prove the existence of certain mathematical objects, such as choice functions, that are necessary for many mathematical constructions and proofs.

5. What are some criticisms of the Axiom of Choice?

One of the main criticisms of the Axiom of Choice is that it can lead to counterintuitive results, such as the Banach-Tarski paradox, which states that a solid ball can be decomposed into a finite number of pieces and reassembled to form two identical copies of the original ball. Additionally, some mathematicians argue that the Axiom of Choice is not necessary for most mathematical proofs and that its use can be avoided by carefully formulating theorems and definitions.