[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.

[jacksnipe]

It’s not really considered throwing away Peano’s work. Peano was working in the very infancy of logical formalism.

As it turns out, further work developing on his discovered that using a recursively enumerable schema for induction rather than a second-order induction axiom gives rise to a simpler abstraction that still has all the properties that Peano actually desired, and which makes further developments in the space much easier.

Continuing to call it Peano Arithmetic is respect for the fact that the guy got it mostly right, and it took the mathematics community many more years to refine the ideas to their current point.

Is it a shame that Galois theory isn’t presented as a historical fossil and frozen to its state of development in Galois’s lifetime? I may be making a rather big assumption, but I like to think he would be proud, and so would Peano.

[JadeNB]

> Continuing to call it Peano Arithmetic is respect for the fact that the guy got it mostly right, and it took the mathematics community many more years to refine the ideas to their current point.

> Is it a shame that Galois theory isn’t presented as a historical fossil and frozen to its state of development in Galois’s lifetime? I may be making a rather big assumption, but I like to think he would be proud, and so would Peano.

Oh, by no means do I object to calling an updated and generalised version by the name of the person who originated the subject! Since you've brought up Galois, I hardly think that he'd recognize the modern theory of Galois connections, but I think that the name is wholly appropriate.

No, what I thought was a shame is if the original theory doesn't get discussed at all. If my only exposure to Peano's work was, for example, the axiom schema in Enderton, then I don't think I'd be able to appreciate why it's such a big deal. That would feel to me like teaching the theory of Galois connections without ever saying anything about field theory! Whereas, on the other hand, I did immediately understand as an undergraduate the magic of being able to define everything in terms of the pointed set using induction, and I think I'd appreciate even more having seen that first and then seeing how it is updated for modern mathematical logic.

In fact, at a casual glance, I still don't see why L1, L3, and the A, M, and E axioms can't be omitted in the presence of the axiom(s) on p. 269, which has been the whole substance of my objection. I believe that there's an answer, but, if I don't see it as a professional mathematician (though not a logician), then surely it can't be true that every undergraduate will appreciate it!

[jacksnipe]

Second addendum, chapter 4 is about second order logic and apparently I just forgot that exercise 1 is simply showing that you get all of the structure built up in Chapter 3 with Peano's original formulation in second-order logic. Seems that here I'm the one suffering from a lack of historical context!

I think from a logic standpoint this also makes sense -- getting to undecidability quickly makes taking the direct route through first-order logic more appealing.

If I'm being honest, I now do feel a little bit deprived, I probably would have enjoyed the categorical view when I was learning this too.

[jacksnipe]

Oh, then yeah, I totally agree. In general I think it's a shame that so little emphasis is placed on the history of mathematics, though at the same time I appreciate that most of my peers just didn't care :(

[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.)