The axiom of choice is a fundamental principle in mathematics that allows us to make choices from a collection of sets, even if the collection is infinite. It is one of the most controversial and debated topics in mathematics, and has been the subject of much discussion and research over the years.

## Definitions

The axiom of choice is a statement in set theory that asserts that for any collection of non-empty sets, there exists a way to choose one element from each set. In other words, it states that given a collection of sets, we can always choose one element from each set to form a new set.

## Origin

The axiom of choice was first introduced by the German mathematician Ernst Zermelo in 1904 as a means of proving the well-ordering theorem. However, the axiom of choice was not widely accepted at the time, and it was not until the 1920s that it became more widely recognized as a fundamental principle of mathematics.

## Meaning in different dictionaries

According to the Oxford English Dictionary, the axiom of choice is “a principle of set theory stating that given any collection of non-empty sets, it is possible to choose one element from each set.” The Merriam-Webster Dictionary defines it as “a statement in set theory that asserts the existence of a selection from any set of mutually disjoint nonempty sets.”

## Associations

The axiom of choice is closely related to other fundamental principles in mathematics, such as the well-ordering theorem, the Zorn’s lemma, and the Hahn-Banach theorem. It is also used in many areas of mathematics, including topology, analysis, and algebra.

## Synonyms

There are no commonly used synonyms for the axiom of choice.

## Antonyms

There are no commonly used antonyms for the axiom of choice.

## The same root words

The term “axiom” comes from the Greek word “axios,” meaning “worthy.” The word “choice” comes from the Old English word “cēosan,” meaning “to choose.”

## Example Sentences

- The axiom of choice is a fundamental principle in mathematics that allows us to make choices from a collection of sets.
- Without the axiom of choice, many important results in mathematics would not be possible.
- The axiom of choice is a controversial topic in mathematics, with many mathematicians debating its validity.
- The axiom of choice is closely related to other fundamental principles in mathematics, such as the well-ordering theorem and the Hahn-Banach theorem.
- The axiom of choice is used in many areas of mathematics, including topology, analysis, and algebra.