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anton@lemmy.blahaj.zone ⁨7⁩ ⁨hours⁩ ago

But I am not taking about an amount of different things, but a parallel or branching number line being part of the set of natural numbers.
I am not talking about modular arithmetic on its own, but as part of the set of natural numbers.

Under the missing axioms those constructs would be part of the natural numbers, including an x in N such that s(x)=x and therefore x+1=x. While some might think this implies 0=1, it doesn’t, because we don’t have the axiom of induction, an thus can’t prove a+c=b+c => a=b.

The usefulness of such a system questionable but it certainly doesn’t describe the natural numbers as we understand them.

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