Counting on is the same as the right additive identities.
Counting on by zero from a number is the number.
Counting on by one more from the same anchor number is counting on by one more from the prior sum.
So 3+2′ = (3+2)’
Here 3 is the anchor and 2 and 2′ are the counting on indicators. I call these shifts in my book on Peano Axioms.
Counting on from an anchor forever will generate the tail of whole numbers that starts from that number. This is the smallest set that contains 2 and is closed under counting on.
We encounter the intermediate sums as we count on by one successively. This is the same as the recursion equation above.
To count on by 2′, we first count on by 2. So we encounter 5 before we encounter 5′.