Instead of saying induction, we can say infection or even just spreads. If 0 is infected, and it spreads to the next number, then all the numbers are infected.
Infected can mean some proposition is true. The proposition can be membership in a set.
In definition by recursion, infected means defined. So addition is defined by the right additive identities as
x+0 = x
x+y’ = (x+y)’
So we start at x as the offset for the count. But we count the y, what we add, starting from 0.
x+0, means x is infected. That is addition is defined for y=0. x+y is defined to be x at y=0.
Suppose that x+y is infected, so x+y is defined for y, holding x fixed. Then x+y’ is infected, and it is infected as (x+y)’.
So addition spreads in the y variable, the second addend. We can call x the offset and y the increment. So for a given offset, the sum spreads to neighboring y values, with the sum incrementing by one as the y increments by one.