I have what I hope are some simple questions regarding the Peano axioms. I am reading Terence Tao's book "Analysis 1" where he constructs the natural numbers using the Peano axioms, but the axioms are written in plain English. I want to convert these axioms into FOL sentences (and SOL sentence for the axiom of induction).
However, I can't find a complete and consistent list of these axioms in this form, so I have pieced together from various sources how I think they should be:
Signature: $\mathcal{L} = \{\mathbb{N}, 0, S\}$
- $\exists x(x = 0 \wedge x \in \mathbb{N})$
- $\forall n(n \in \mathbb{N} \to S(n) \in \mathbb{N})$
- $\forall n(S(n) \ne 0)$
- $\forall m \forall n(S(m) = S(n) \to m = n)$
- $\forall P(P(0) \wedge (P(n) \to P(S(n))) \to \forall n P(n))$
First question: Is this a correct way of writing the axioms in logic?
Second question: The signature for the Peano axioms doesn't include the $\in$ symbol as a non-logical symbol in any of the treatments I've encountered on the web. Does this mean that the membership predicate is in some way "built-in" to second-order logic so that you don't need to declare it in the signature? To me this makes sense, since SOL can quantify over sets of objects, so therefore the notion of what a set is and what membership is, is integral to SOL.