Banach-Tarski and Dark Energy

18 “Bring them here to me,” he said. 19 And he directed the people to sit down on the grass. Taking the five loaves and the two fish and looking up to heaven, he gave thanks and broke the loaves. Then he gave them to the disciples, and the disciples gave them to the people. 20 They all ate and were satisfied, and the disciples picked up twelve basketfuls of broken pieces that were left over. 21 The number of those who ate was about five thousand men, besides women and children....Matthew 14: 18-21

The Banach-Tarski Theorem is the mathematical equivalent of a free lunch. It says that with the aid of a natural appearing and frequently indispensable assumption about set theory (The Axiom of Choice), one can decompose a solid sphere into component parts and reassemble them to make two spheres each identical to the original. I have no reason to believe that it has anything to do with dark energy except for that free lunch bit.

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