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( S i ) i I {\displaystyle (S_{i})_{i\in I}} of nonempty sets there exists an indexed family ( x i ) i I {\displaystyle (x_{i})_{i\in I}} of elements such that x i S i {\displaystyle x_{i}\in S_{i}} for every i I {\displaystyle i\in I} . 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. Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in the standard form of axiomatic set theory, ZermeloFraenkel set theory with the axiom of choice (ZFC). One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy. The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

During his reign, he was one of the most feared enemies of the Western and Eastern Roman Empires. He crossed the Danube twice and plundered the Balkans, but was unable to take Constantinople. His unsuccessful campaign in Persia was followed in 441 by an invasion of the Eastern Roman (Byzantine) Empire, the success of which emboldened Attila to invade the West. He also attempted to conquer Roman Gaul (modern France), crossing the Rhine in 451 and marching as far as Aurelianum (Orlans) before being defeated at the Battle of the Catalaunian Plains. He subsequently invaded Italy, devastating the northern provinces, but was unable to take Rome. He planned for further campaigns against the Romans but died in 453. After Attila's death his close adviser Ardaric of the Gepids led a Germanic revolt against Hunnic rule, after which the Hunnic Empire quickly collapsed.