[JadeNB]

> Zero is a natural number. It is in the axioms of Peano arithmetic, and any other definition is just teachers choosing a taxonomy that best fits their lesson.

It is, but it need not be. In the category of pointed sets with endofunctor, (Z_{\ge 1}, 1, ++) and (Z_{\ge 0}, 0, ++) are isomorphic (to each other, to (Z_{\ge 937}, 937, ++), and to any number of other absurd models), so either would do equally well as a model of Peano arithmetic.

[MITSardine]

I may be misunderstanding your argument, but if it's that of a simple offset, then only the one starting from 0 forms a monoid (a group without an inverse to each element). Though, of course, you could redefine the + operation...

[JadeNB]

> I may be misunderstanding your argument, but if it's that of a simple offset, then only the one starting from 0 forms a monoid (a group without an inverse to each element). Though, of course, you could redefine the + operation...

Yes, agreed, there is other algebraic structure that can tell the difference, but Peano arithmetic by itself cannot.

[jacksnipe]

I think I’m missing something here. PA defines x * 0 = 0 for all x. So while we could take (Z+, 1, ++) as a model of it, we would be imposing a completely different definition of multiplication than the usual. Would this not be simply choosing to label 1 as 0 and work from there?

[JadeNB]

> I think I’m missing something here. PA defines x * 0 = 0 for all x. So while we could take (Z+, 1, ++) as a model of it, we would be imposing a completely different definition of multiplication than the usual. Would this not be simply choosing to label 1 as 0 and work from there?

Despite the name, in the usual mathematical meaning of the term, Peano arithmetic does not define arithmetic at all, only the successor operation, and everything else is built from there. Once we have those, for the model (Z_{\ge 0}, 0, ++), we certainly usually do define x0 = 0 for all x; and, you're right, if for the model (Z_{\ge 1}, 1, ++) we defined x1 = 1 for all x (as no-one could stop us from doing), then we'd just be dealing with "0 by another name." But it might be equally sensible, if our model of Peano arithmetic is (Z_{\ge 1}, 1, ++), to define x1 = x for all x, in which case we recover the expected arithmetic.

[jacksnipe]

Sorry, but that’s incorrect. Multiplication is defined in the Peano axioms, in terms of S(x).

2 of the axioms are:

1. For all x, x*0 = 0

2. For all x, y: x*S(y) = x*y + y

[JadeNB]

In the usual terminology, these are not axioms; as your wording itself says, they are definitions. (Indeed, I'd argue that it's almost ungrammatical to say something is "defined in the axioms"; axioms may, and probably must, be stated in terms of definitions, but the definitions are not themselves axioms.) As I say, one can quibble about terminology, since what's important is less what's axiom and what's definition, and more what we can build on top of both; but the usual mathematical presentation separates out the axioms (numbered 1–9 at https://en.wikipedia.org/wiki/Peano_axioms#Historical_second..., though things like 2–5 wouldn't usually be stated as an axiom of the theory but rather of the ambient logic) from the definitions (see https://en.wikipedia.org/wiki/Peano_axioms#Defining_arithmet...).

(Now having written that and looking back, I see that, in my previous post https://news.ycombinator.com/item?id=43442074, I wrote "Despite the name, in the usual mathematical meaning of the term, Peano arithmetic does not define arithmetic at all, only the successor operation, and everything else is built from there." Perhaps this infelicitious-to-the-point-of-wrong wording of mine is the source of our difference? I meant to say that Peano arithmetic does not axiomatize arithmetic at all, but that arithmetic can be defined from the axioms. Thus the specific definition x[pt] = [pt] is eminently sensible if we consider the distinguished point [pt] to be playing the usual role of 0; but the definition x[pt] = x is also sensible if we consider it to be playing the usual role of 1, and even things like x[pt] = x + x + x + x + x can be tolerated if we think of [pt] as standing for 5, say. The axioms cannot distinguish among these options, because the axioms say nothing about multiplication.)

[jacksnipe]

No, they are axioms. Peano arithmetic itself is a first-order theory, and a theory is just a recursively enumerable set of axioms.

Enderton, “A Mathematical Introduction to Logic, 2nd Ed.”, p,203,269-270

Kleene, “Mathematical Logic”, p.206

EDIT: It seems like you're talking about Peano's original historical formulation of arithmetic? That's all well and good but it is categorically not what is meant by "Peano Arithmetic" in any modern context. I've provided two citations from pretty far apart in time editions of common logic texts (well, "Mathematical Logic" is a bit of a weird book, but Kleene is certainly an authority) and I hope that demonstrates this.

There's a lot of reasons that the theory is pretty much always discussed as a first-order theory. The biggest, of course, is that when taken as a first-order theory it fits neatly into the proof and statement of Godel's Incompleteness Theorems, but iiuc it's just generally much less useful in a model theoretic context to take it as a second order theory (to the point where I only ever saw this discussed as a historical note, not as a mathematical one).

EDIT 2: This is all a digression anyway. Both first- and second-order PA label the start of the Z-chain as 0; so any model of PA contains 0 when interpreted as a model of PA.

[JadeNB]

I'm away from my library, but fortunately the books you referenced are a Google away, so I could consult them and confirm that they say what you say. I'm not quite willing to accept Kleene's word as an authority on common modern mathematical practice, since he was a theoretical computer scientist before the term, but, though I'm not familiar with Enderton's book, it certainly looks like a reasonably standard one.

But these are all referring to Peano arithmetic as a model of the theory of the natural numbers. And that seems a bit silly: the impact of Peano's work wasn't because he showed that there was a model of the theory of the natural numbers, which everybody believed if they bothered to think about it, but because he showed that all you needed to make such a model was a successor operation satisfying certain axioms. Yes, they may be less model-theoretically congenial because they're second order, but to change Peano's work from what he did historically and still call it Peano's seems strange to me. (I'm fine with dressing it up in modern language, and calling it an initial object in the category of pointed sets with endofunctor, which perhaps is biased but still seems to me to be capturing the essential idea.)

Certainly I was taught the second-order approach, though it was as an undergraduate; I've never taken a model-theory class. As I say, I'm away from my library and so can't consult any other sources to see if they still teach it this way, and anyway I am a representation theorist rather than a logician; but, if the common logical approach these days really is to discard Peano's historical theory and to call by Peano's name something that isn't his work, even if it is more convenient to use, then I think that's a shame from the point of view of appreciating the novelty and ingenuity of his ideas. But just because I think something is a shame doesn't mean it's not true, and so far you've produced evidence for your view and I can't for mine, so I can't argue any further.

[JadeNB]

> EDIT 2: This is all a digression anyway. Both first- and second-order PA label the start of the Z-chain as 0; so any model of PA contains 0 when interpreted as a model of PA.

Ah, good point that this was the actual source o# the discussion. This one at least can be argued, because the question is about how things should be axiomatized/defined, not how they are. And certainly the theory of the "natural numbers starting with 1" can be axiomatised just as well as the "natural numbers starting with 0." All these axioms are made by humans, and an appeal to existing axioms here can only say what's been done, not what should be. (And I say this as someone who does start my naturals at 0.)