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I think you may have not gotten far enough to get a formal explanation of what 0 and 1 are.

The definition taught to most mathematicians is that for the natural numbers,

  0 = {}
  1 = {{}}
In general

  S(N) = N ∪ {N}
But this is not how the real numbers are constructed. You either use Dedekind cuts, Cauchy sequences, or something else.


Complex numbers are just pairs (real and imaginary) of Cauchy sequences of pairs (numerator and denominator) of pairs (positive and negative) of nested versions of the empty set. (modulo al the necesary equivalence to make this work).

So the natural number {{}} is canonical included in the complex numbers as [something].

Natural {{}}

Integer ({{}},{})

Rational (({{}},{}),({{}},{}))

Real (({{}},{}),({{}},{})), (({{}},{}),({{}},{})), (({{}},{}),({{}},{})), ...

Complex ((({{}},{}),({{}},{})), (({{}},{}),({{}},{})), (({{}},{}),({{}},{})), ... , (({},{}),({{}},{})), (({},{}),({{}},{})), (({},{}),({{}},{})), ... )

(And there are a few alternative definition, but I think they are even more messy.)




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