I was watching a set of lectures on introductory real analysis. [LINK](https://www.youtube.com/watch?v=Nh57TcGJelU&list=PLo4jXE-LdDTRh_XIyhRwyCup4J6D8ftTL&index=1). Brazilian portuguese only unfortunately.

The lecturer starts defining (his own words):

Peano's axioms
==============

Let $\mathbb{N}={1,2,3,\dots}$:

1. There exists an injective function $s:\mathbb{N}\rightarrow\mathbb{N}$;
2. There exists an element $1$ such that $1 \in \mathbb{N}$ but $1 \notin s(\mathbb{N})$, i.e. $1 \notin Im(s)$;
3. (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:

1. 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).