Counting on by one from a whole number is the same as the successor function. We use the same notation, prime.
We can use S for successor
S(2) = 3
If we used C for counting on by one, then
C(2) = 3
If we use C(i,j) for counting on from i by j, then
C(i,0) = i
C(i,j’) = C(i,j)’
These are the same identities as for addition.