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thomblake 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.
Sure there is, as long as you realize that "give" and "take" are the same action. Giving -1 apples is just taking 1 apple.
Number of apples isn't closed under taking.
If you wish to satisfy the ring axioms from scratch, you must first invent the economy...