Mmm, long division is an interesting one. Who am I to say how things must be
proved, but the proofs of the division algorithm with which I am familiar
involve the well-ordering principle. There, in this one idea, lies two
problematic details:
1. The non-algebraic nature of the well-ordering principle
<https://en.wikipedia.org/wiki/Well-ordering_principle> , and its
correlative controversies . As outlined in the paper, "It has been shown
that if you want to believethe well-ordering theorem, then it must be taken
as an axiom."
2. The first significant moment where intension in the form of
computational complexity enters an otherwise extensional number theory.
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