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 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.   

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

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