* Arithmetic in CC cats, query
@ 1999-11-06 1:41 Colin McLarty
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From: Colin McLarty @ 1999-11-06 1:41 UTC (permalink / raw)
To: categories
Take cartesian closed categories in the sense: having products, and
internal homs. If there is a natural number object you can define addition
NxN-->N with the usual recursion
0+n = n sm+n = s(n+m)
But can you prove it is commutative? You can prove 0+n=n+0 because you can
prove both = n. And you can prove each case of commutativity for "standard
natural numbers" in the sense of global elements sss..ss0 gotten from 0 by
(externally) finitely many applications of successor.
But even if I also assume equalizers, I do not see how to prove
commutativity for arbitrary global elements, let alone all generalized
elements. I suspect there are counterexamples but I cannot give one.
I wonder if this is in Lambek and Scott, but my copy is at the office and
I will not be there until Monday so I am writing in to ask.
best, Colin
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1999-11-06 1:41 Arithmetic in CC cats, query Colin McLarty
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