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In every model of a theory, you can just consistently relabel elements in order to get another model. But that doesn't really tell you anything because that mod

In every model of a theory, you can just consistently relabel elements in order to get another model. But that doesn't really tell you anything because that model is just isomorphic to the old one (that term has a precise meaning, but I think the "relabeling" intuition captures it). So your NPA is just a consistent relabeling of elements n to elements -n - it's fully isomorphic to PA. There's nothing particularly interesting about it. Notice that "the product of two negative numbers is positive" doesn't help because in your alternative model, "positive" (or rather, "nonnegative") just means something different (or more precisely, it doesn't even make sense to talk about "nonnegative numbers" because that's all of them). It only becomes interesting when you have different non-isomorphic models of a theory - that then means that there are (distinguishable) properties of the model that the theory can't talk about. In the case of the complex numbers, if you just exchange i and -i, I believe that you get a field which is fully isomorphic to C.

Tainnor 2y ago
> In the case of the complex numbers, if you just exchange i and -i, I believe that you get a field which is fully isomorphic to C. To add to this and expand a little, there are only two isomorphisms between C and itself which keep all the real numbers fixed: the identity and the conjugate map (which sends a+ib to a-ib, and in particular i to -i). If you don't fix the real numbers there are other such isomorphisms but I believe you can only construct them via the axiom of choice (so you can't really "write them down"). In the case of the real numbers there's really only one such isomorphism, the identity.
- lisper 2y ago
  Right. So in those terms, NPA is isomorphic to PA -- until you introduce multiplication, at which point the isomorphism goes away becauase S(0)S(0) is S(0) but P(0)P(0) is not P(0). That seems like an interesting fact to me, and feels like it might be a clue pointing to some deeper truth.
  - Tainnor 2y ago
    You can't define a model "NPA" in which all numbers are nonpositive but then define multiplication such that it can yield positive numbers - that just wouldn't be a function. It's pretty easy to prove that S(0)*S(0) = S(0) with the axioms of PA. That has to be true no matter what interpretation you give "S" in your model, so in your proposed model "NPA", in which the domain are the nonpositive integers and S is the predecessor function, of course it has to be true that the "product" of two -1 and -1 is -1. It just means that in such a model, multiplication isn't what you think it is. One should really be careful not to mix up logical systems and their models. The way you're using notation, it's really easy to get confused.

Tainnor 2y ago

> In the case of the complex numbers, if you just exchange i and -i, I believe that you get a field which is fully isomorphic to C. To add to this and expand a little, there are only two isomorphisms between C and itself which keep all the real numbers fixed: the identity and the conjugate map (which sends a+ib to a-ib, and in particular i to -i). If you don't fix the real numbers there are other such isomorphisms but I believe you can only construct them via the axiom of choice (so you can't really "write them down"). In the case of the real numbers there's really only one such isomorphism, the identity.

lisper 2y ago
Right. So in those terms, NPA is isomorphic to PA -- until you introduce multiplication, at which point the isomorphism goes away becauase S(0)S(0) is S(0) but P(0)P(0) is not P(0). That seems like an interesting fact to me, and feels like it might be a clue pointing to some deeper truth.
- Tainnor 2y ago
  You can't define a model "NPA" in which all numbers are nonpositive but then define multiplication such that it can yield positive numbers - that just wouldn't be a function. It's pretty easy to prove that S(0)*S(0) = S(0) with the axioms of PA. That has to be true no matter what interpretation you give "S" in your model, so in your proposed model "NPA", in which the domain are the nonpositive integers and S is the predecessor function, of course it has to be true that the "product" of two -1 and -1 is -1. It just means that in such a model, multiplication isn't what you think it is. One should really be careful not to mix up logical systems and their models. The way you're using notation, it's really easy to get confused.