this post was submitted on 01 Oct 2025
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Presumably you were starting with a fundamental axiom such as 1 + 1 = 2, which is the difficult one to prove because it's so fundamental
I find this axiomatization of the naturals quite neat:
Now the neat part: If 0 is a constant, then s(0) is also a constant. So we can invent a name for that constant and call it “1.” Now s(s(0)) is a constant, too. Call it “2” and proceed to invent the natural numbers.
That axiomisation is incomplete as it doesn't preclude stuff like loops, a predecessor to zero or a second number line.
There are non-standard models of arithmetic. They follow the original first-order Peano axioms and any theorem about the naturals is true for them, but they have some wacky extra stuff in them like you mention.