A question about the Peano's axioms I was watching a set of lectures on introductory real analysis. LINK. Brazilian portuguese only unfortunately.
The lecturer starts defining (his own words):
Peano's axioms
Let $\mathbb{N}={1,2,3,\dots}$:

*

*There exists an injective function  $s:\mathbb{N}\rightarrow\mathbb{N}$;

*There exists an element $1$ such that $1 \in \mathbb{N}$ but $1 \notin s(\mathbb{N})$, i.e. $1 \notin Im(s)$;

*(induction) Let $A$ be a set such that $A \subset \mathbb{N}$. If $1 \in A$ and ($n \in A \implies s(n) \in A$) then $A = \mathbb{N}$.

I'm still learning the very (very) basics of this, so I'm not sure whether this a proper way to define this stuff neither I can tell if these are right or wrong.
The question that bugs me is:

*

*He assumed that there is such a thing as a "function" and also assumed "advanced" concepts like "injection". Can he do that given we are in the terrain of very primitive mathematical foundation, i.e. things that all other things build upon in math? In my beginner's math mind he should have used more primitive mathematical concepts (I'm not sure which ones exactly).

 A: As others have pointed out, things like "injective function" can be rephrased in more elementary way. The induction axiom is where all the action is. In order to make sense of the induction axiom, we need to have a notion of arbitrary subsets of $\mathbb N,$ which requires some set theoretical commitments. In other words, we need second-order and not merely first-order logic to state and interpret this.
One way to "get around" this is to replace the induction axiom with a first-order schema. So we have for each first-order formula $\varphi(x,\vec y)$, a first-order induction axiom Ind$_\varphi$: $$ \forall \vec y\;((\varphi(0,\vec y)\land \forall x\;(\varphi(x,\vec y)\to \varphi(Sx,\vec y)))\to \forall x\;\varphi(x,\vec y)).$$
Now our theory is first-order and we can make sense of it without any set theory. However, it's also much weaker and does not determine the natural numbers up to isomorphism (in other words, it has nonstandard models, which is pretty much inevitable with purely first-order axiomatizations).
In my experience "Peano Axioms" refers to the five axioms in the second-order formulation. Whereas "Peano Arithmetic" (aka PA) refers to a first-order axiomatization with an infinite schema of induction axioms that also includes the arithmetical operations $+$ and $\cdot$ in the language.
A: Peano arithmetic is usually defined in a way to avoid this. It is not hard to change. Instead of defining successor as a function, you say "Every integer has a successor." And instead of saying it is an injective function, you say "If $n, m$ have the same successor, then $n = m$."
A: I like very much the way Terence Tao describes the Peano's axioms.
Complementing the answer given by @MichaelBarz and the comments of @ThomasAndrews, I advise you to consider the following formulation:

*

*$0$ is a natural number.

*Every successor of a natural number is a natural number.

*$0$ is not the successor of any natural number.

*Different natural numbers have different successors.

*Induction axiom.

In his book "Analysis I", he discusses the motivation for each of them. It is very enlightening.
Hopefully this helps!
