Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. It states that for every indexed family of nonempty sets there exists an indexed family of elements such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin. In many cases such a selection can be made without invoking the axiom of choice; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin. To give an informal example, for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks (assumed to have no distinguishing features), such a selection can be obtained only by invoking the axiom of choice.

Axiom of choice (disambiguation)

Axiom of choice may refer to:

Axiom of Choice (band)

Axiom of Choice is a southern California (USA) based world music group of Iranian émigrés who perform a fusion style incorporating Persian classical music and modern Western musical concepts.

History

Led by Loga Ramin Torkian, who plays a variant of a guitar of his own invention that is fretted to play quarter tones, the band has a sound combining soaring female vocals, Middle-Eastern rhythms and melodies, and progressive Western production styles.

The band was named after the mathematical concept, the axiom of choice.

The melodies and rhythms of Persia's radif tradition are mixed with Western music. Led by Persia-born nylon-string classical guitar, quarter-tone guitar, and tarbass player and musical director Loga Ramin Torkian, the septet incorporates a global range of influences into its sound. Axiom of Choice remains rooted in the musical heritage of Torkian and fellow Persian émigrés Mamak Khadem and Pejman Hadadi. While Khadem's singing in the Persian language retains the spirit of the past, the playing of Hadadi, one of the leading Persian percussionists living in the United States, gives the group its flavor. Hadadi—who plays daf, tombak, and nagada—previously toured with Hossein Alizadeh, Kayhan Kalhor, and Shahram Nazeri. He continues to be involved with Axiom of Choice and with the Dastan ensemble.