How does Peano’s recursive definition of addition calculate sums of natural numbers?

Peano’s recursive definition of addition provides a formal, step-by-step method to calculate the sum of two natural numbers using the successor function, denoted as $S(n)$, which represents the next natural number after $n$.12

The definition relies on two fundamental rules:
* $x+0=x$
* $x+S(y)=S(x+y)$

These rules allow for the calculation of any sum by systematically reducing the second number until it reaches zero. For example, to calculate $3+2$, where $2$ is defined as $S(1)$ and $1$ as $S(0)$, the process unfolds as follows:

$3+2=3+S(1)=S(3+1)$
$S(3+1)=S(3+S(0))=S(S(3+0))$
Since $3+0=3$, the expression simplifies to $S(S(3))$, which is $S(4)$, resulting in $5$.2

By applying these recursive steps, complex arithmetic operations are reduced to a finite sequence of successor applications, ensuring that addition is rigorously defined within the Peano axioms framework.23