Kronecker famously wrote, “God created the natural numbers; all else is the work of man”. The truth of this statement (literal or otherwise) is debatable; but one can certainly view the other standard number systems $latex {{bf Z}, {bf Q}, {bf R}, {bf C}}&fg=000000$ as (iterated) completions of the natural numbers $latex {{bf N}}&fg=000000$ in various senses. For instance:

- The integers $latex {{bf Z}}&fg=000000$ are the additive completion of the natural numbers $latex {{bf N}}&fg=000000$ (the minimal additive group that contains a copy of $latex {{bf N}}&fg=000000$).
- The rationals $latex {{bf Q}}&fg=000000$ are the multiplicative completion of the integers $latex {{bf Z}}&fg=000000$ (the minimal field that contains a copy of $latex {{bf Z}}&fg=000000$).
- The reals $latex {{bf R}}&fg=000000$ are the metric completion of the rationals $latex {{bf Q}}&fg=000000$ (the minimal complete metric space that contains a copy of $latex {{bf Q}}&fg=000000$).
- The complex numbers $latex {{bf C}}&fg=000000$ are the algebraic…

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