139
edits
Changes
Jump to navigation
Jump to search
Line 26:
Line 26: − − The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians. {{fact}}
remove unsource statement
Use of the Axiom of Choice has led to some seemingly absurd results. In the [[Banach-Tarski Paradox]], the Axiom of Choice is used to prove that a solid sphere of infinitely divisible parts may be chopped up and reconstructed as two new spheres of identical size, thereby creating 2 out of only 1. This paradox is proven only through use of the Axiom of Choice, and the authors of this proof did so to criticize this Axiom. One attempt to resolve this apparent contradiction is to show that physical spheres are not [[Lebesgue measurable]].<ref>http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>
Use of the Axiom of Choice has led to some seemingly absurd results. In the [[Banach-Tarski Paradox]], the Axiom of Choice is used to prove that a solid sphere of infinitely divisible parts may be chopped up and reconstructed as two new spheres of identical size, thereby creating 2 out of only 1. This paradox is proven only through use of the Axiom of Choice, and the authors of this proof did so to criticize this Axiom. One attempt to resolve this apparent contradiction is to show that physical spheres are not [[Lebesgue measurable]].<ref>http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>
== Use of the Axiom of Choice ==
== Use of the Axiom of Choice ==