I am confused about how we should state the Peano axioms

Question

I am really confused if the Peano axioms are supposed to be strictly a set theory / first order thing or how are we supposed to state them. In English? Can we use purely logical expressions? What about induction? Does this correctly state the axioms?

1. $0 \in \mathbb{N}$

2. $\forall a \in \mathbb{N}, S(a) \in \mathbb{N}$

3. $\forall a \in \mathbb{N}, S(a) \neq 0$

4. $\forall a, b \in \mathbb{N}, S(a) = S(b) \to a = b$

5. $\forall P(P(0) \land \forall k(P(k) \to P(k+1)) \to \forall n (P(n)))$

That fifth one I am pretty sure is second order logic, which I don't know if that's the right way to do it or not. Not sure if the correct one is some first-order thing instead.

Are these the correct way to state axioms? I see so many different formulations and a lot of them are just in English, is that the way it's normally stated, informally like that, or is there a formal and standard way to say "these are the Peano axioms for defining how natural numbers work"?

Answer 1

The Peano axioms are a first-order theory in which the only objects are natural numbers (in this context defined to start at $0$, not $1$), so you can't have $\Bbb N$ suddenly show up. A better statement of your axioms would be:

1. $\exists 0\forall a (a+0=a)$
2. $\forall a\exists b(Sa=b)$
3. This one's fine as-is for now
4. This one too
5. Add one axiom per choice of $P$, rather than quantifying over $P$ (that's as forbidden as having $\Bbb N$ appear); for better or worse, we need infinitely many axioms.

But if you want to really go to town, the axioms here add a lot more. They break down the rules for $=$ and $+$, and even $\times$ (because we're not just doing Presburger arithmetic here, so e.g. you need $aSb=ab+a$). In particular, the axioms for $+,\,\times$ provide more information about $S$.

Answer 2

The axioms you've written are essentially those of Peano. He worked in an informal set theory, and began with an axiom saying that $\mathbb N$ is a set (or a class --- the distinction didn't yet exist). Also, he started his $\mathbb N$ with $1$, not $0$. So, from a historical point of view, your axioms have a reasonable right to be called the Peano axioms.

Instead of set theory, one can base these axioms on second-order logic. In particular, second-order logic allows the quantification over predicates, $\forall P$, in the induction axiom.

More recently, though, the phrase "Peano arithmetic" (with the abbreviation PA) has come to be used almost universally for a first-order approximation to these second-order axioms --- replacing the second-order induction axiom, which refers to all predicates, with a schema of first-order axioms (as in J.G.'s answer), referring just to the (parametrically) definable predicates.

There are major differences between the original second-order axioms and the first-order theory PA. In particular, the second-order axioms admit only one model up to isomorphism, namely the standard natural numbers. In contrast, PA, like any first-order theory with an infinite model, admits many non-isomorphic models.