https://secure.wikimedia.org/wikipedia/en/wiki/Banach%E2%80%93Tarski_paradox https://secure.wikimedia.org/wikipedia/en/wiki/Banach%E2%80%... It works because,

https://secure.wikimedia.org/wikipedia/en/wiki/Banach%E2%80%93Tarski_paradox https://secure.wikimedia.org/wikipedia/en/wiki/Banach%E2%80%... It works because, if you assume the axiom of choice, you can postulate "pathological" sets that don't have a mathematically well-defined volume. You can use this to backdoor volume changes, by cutting something with well-defined volume into a buncha pieces without a well-defined volume, then reassembling this into something with a different volume than the original. You can't get around this trick by redefining what it means to measure something: If you pick any reasonable definition, it must admit the Banach-Tarski paradox. https://secure.wikimedia.org/wikipedia/en/wiki/Non-measurable_set https://secure.wikimedia.org/wikipedia/en/wiki/Non-measurabl... These pathological sets are too weird to exist in physical reality.