• TeddE@lemmy.world
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    16 hours ago

    Not sure what you mean by ‘loops’ - except perhaps modular arithmetic, but there are no natural numbers that are negative - you may be thinking of integers, which is constructed from the natural numbers. Similarly, rational numbers, real numbers, and complex numbers are also constructed from the naturals. Complex numbers are often expressed as though they’re two dimensional, since the imaginary part cannot be properly reduced, e.g. 3+2i.

    I recommend this playlist by mathematician another roof: https://www.youtube.com/playlist?list=PLsdeQ7TnWVm_EQG1rmb34ZBYe5ohrkL3t

    They build the whole modern number system ‘from scratch’

    • anton@lemmy.blahaj.zone
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      9 hours ago

      I know how how natural numbers work, but the axioms in the comment i replied to are not enough to define them.

      Not sure what you mean by ‘loops’

      There could be a number n such that m=s(n) and n=s(m). This would be precluded by taking the axiom of induction or the trichotomy axiom.

      If we only take the latter we can still make a second number line, that runs “parallel” to the “propper number line” like:

      n,s(n),s(s(n)),s(s(s(n))),...
      0,s(0),s(s(0)),s(s(s(0))),...
      

      there are no natural numbers that are negative

      I know, but the given axioms don’t preclude it. Under the peano axioms it’s explicitly spelled out:
      0 is not the successor of any natural number