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JGWeissman comments on Exponentiation goes wrong first - Less Wrong
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For models for which axiom 3 (0 is not a successor of any number) is false, but the others, including induction, hold, look at modular arithmetic.
Consider the set {n / 2 : n is a natural number } which follows all axioms except induction.
I think pointing out that modular arithmetic violates axiom 3 is akin to pointing out that in yahtzee, four-in-a-row beats three-of-a-kind. You're not really disputing the rules of poker. I'm speaking gut-wise, not mathematically here.
The point is that that axiom 3 and induction both exclude some models while allowing others. They are both part of defining the natural numbers.
I can't tell if you're arguing with me. 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 -- in fact I would agree with Nelson and say that they are what people thought of as natural numbers before induction was codified.
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.
There are a lot of systems that look almost like the natural numbers but clearly are not. I claim that even if you replace 5 with a succession of weaker axioms you can still get objects that are clearly not the natural numbers.
Consider for example the much weaker claim that every element in N is either 0 or the successor of some other element. It is still easy to see that one can have things that don't look at all like N. And one can keep adding additional theorems about what N needs to satisfy and still get objects that are pretty concrete and don't look like what we want the natural numbers to satisfy.
You can actually get some other very weird systems if you want to play around. For example, if you have some underlying set theory and replace induction with the well-ordering principle (which is equivalent modulo some minor technicalities) you can then look at some very interesting systems. For example, well-ordering is second-order, and applies to all sets, so this is in some sense stronger than some first order descriptions of PA where you replace induction with a separate induction axiom for each definable predicate. So one obvious system then is to restrict your well-ordering claim so that it quantifies not over all subsets but over all subsets of some form. Visualizing what such weakened versions of N look like can be quite hard.