Changes

The '''Axiom of Choice''' ('''AC''') in [[set theory]] states that "for every [[nonempty set]] there is a function that chooses an element from each set." This axiom is powerful because by assuming the existence of such a function, one can then manipulate the function to prove otherwise unprovable theorems.  This axiom is controversial because it assumes the existence of a function without demonstrating that it exists, or knowing all of its characteristics. This axiom has been used to prove an apparently absurd result, as in the [[Banach-Tarski Paradox]] discussed below.

The '''Axiom of Choice''' ('''AC''') in [[set theory]] states that "for every set made of [[nonempty set]]s there is a function that chooses an element from each set". This function is called a ''choice function''.

 

This axiom is powerful because by assuming the existence of such a function, one can then manipulate the function to prove otherwise unprovable theorems.  This axiom is controversial because it assumes the existence of a function without giving any hint on how it could be constructed. This axiom has been used to prove many apparently absurd results, as in the [[Banach-Tarski Paradox]] discussed below.

To better understand the Axiom of Choice, consider alternative descriptions of it:

To better understand the Axiom of Choice, consider alternative descriptions of it: