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CCC comments on Proofs, Implications, and Models - Less Wrong
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Come to think of it, apples don't actually satisfy the ring axioms. In particular, if you have at least one apple, there is no number of apples I can give you such that you no longer have any apples.
Does an apple composed of antimatter still count as an apple? If I add that to the original apple, I get a big explosion and lots of energy flying around, but neither apple actually remains afterwards.