Peano Axioms.
Axiom 2.1
$0$ is a natural number.
Axiom 2.2
If $n$ is a natural number then $n++$ is also a natural number. (Here $n++$ denotes the successor of $n$ and previously in the book the notational implication has been bijected to the familiar $1,2…$).
Axiom 2.3
$0$ is not the successor of natural number; i.e. we have $n++≠0$ for every natural number $n$.
Axiom 2.4
Different natural numbers must have different successors; i.e., if $n,m$ are natural numbers and $n≠m$, then $n++≠m++$.
Axiom 2.5
Let $P(n)$ be any property pertaining to a natural number $n$. Suppose that $P(0)$ is true, and suppose that whenever $P(n)$ is true, $P(n++)$ is also true. Then $P(n)$ is true for every natural number $n$.
Definition of Addition: Let m be a natural number. We define, $0+m=m$ and suppose we have inductively defined the addtion $n+m$ then we define, $(n++)+m=(n+m)++$. Where $n++$ is the successor of $n$.
Terence Tao has a proof in his Analysis I book, but I couldn't understand . I want a proof line by line like this:
Thm: 3 is a natural number \begin{align*} & 0 \text{ is natural } && \text{Axiom 2.1 }\\ & 0++ = 1 \text{ is natural } && \text{Axiom 2.2}\\ & 1++ = 2 \text{ is natural } && \text{Axiom 2.2}\\ & 2++ = 3 \text{ is natural } && \text{Axiom 2.2} \end{align*}