One of the points of the post is that if you reject axiom 5 you are left with something that strongly resembles the natural numbers
That doesn’t work. If you reject induction, your remaining four axioms also describes the set {(n,m) such that n and m are natural numbers}, where (0, 0) is the zero element and the successor of (n, m) is (n, m + 1). That seems a bit different than the natural numbers, because I added, without violating the first four axioms, additional elements that are not the successor of any element.
I think you’re correct. It is more accurate to say that the counting numbers strongly resemble the natural numbers, than it is to say that every element of every model of axioms 1-4 resembles a natural number.
Note that one can create “weird” models of axioms 1-5 as well.
That doesn’t work. If you reject induction, your remaining four axioms also describes the set {(n,m) such that n and m are natural numbers}, where (0, 0) is the zero element and the successor of (n, m) is (n, m + 1). That seems a bit different than the natural numbers, because I added, without violating the first four axioms, additional elements that are not the successor of any element.
I think you’re correct. It is more accurate to say that the counting numbers strongly resemble the natural numbers, than it is to say that every element of every model of axioms 1-4 resembles a natural number.
Note that one can create “weird” models of axioms 1-5 as well.