EPTOn May 13 2009 22:01 flag wrote:
shouldn't the banach-tarski paradox, which assumes the axiom of choice, disprove the axiom of choice? if you reach a contradiction then one of your assumptions is false. the B-T paradox is a contradiction because after each transformation the volume remains constant, yet after all transformations the volume has doubled.
shouldn't the banach-tarski paradox, which assumes the axiom of choice, disprove the axiom of choice? if you reach a contradiction then one of your assumptions is false. the B-T paradox is a contradiction because after each transformation the volume remains constant, yet after all transformations the volume has doubled.
You assume that the total volume of any two objects (subsets of R^3) should equal the sum of the two volumes. The Banach-Tarski paradox shows that this condition is impossible, there are (very weird) sets that will have different volumes depending on how you assemble them. So it's not a contradiction, it's your conception of volume that is wrong.
However this has nothing to do with objects in the real world, since all objects are made up of finitely many atoms, and all subsets of objects are very "nice" compared to all possible sets of objects in R^3.
