Assuming by degenerate you mean inconsistent, doesn’t proving 1 + 1 = 3 imply your axioms are inconsistent? Unless in your theory you can’t manage to prove that 1 + 1 is anything else...
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In my number theory the number normally referred to as “2” is instead represented by the symbol “3” and vice versa... in all other respects it’s the same
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"the whole is greater than the sum of its parts"
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It would be true in any quotient algebra where 2 is congruent to 3, I guess.
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addition of gray codes. 0x1+0x1=0x11. addition only has to be associative and commutative - under different fields you'll get wildly different answers, just like how 255+1=0 under GP(2^8)
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This is a great answer but/and gray codes are a different way of writing the natural numbers so of course 0x11 == 2 I like it though, it reminds me of ‘equations’ where you ‘cancel’ numbers from a division and accidentally get the right answer. Strange loops are fun!
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I mean if it was less than two it could be trivially done with vector math. Not sure how to get more than two from 1 + 1 without some really absurd hax, like asserting unspecified units were present all along. "One pair and one single is three," that sort of thing.
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Or "the symbols are arbitrary so we can imagine a system of writing numbers in which 3 is this many [ | | ] and 2 is this many [ | | | | ] and 4 is this many [ | | | ]" Or "1 + 1 = 3 because I said so and I'll keep beating you until you agree"
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what do you have against degenerate axioms
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trivial is boring
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