Counting back by one we can denote by ‘n.
So 2 = ‘3.
1 = ”3
Counting back leads us to subtraction.
x-0 = x
x-y’ = ‘(x-y)
The latter equation says if x is our anchor, then counting back one more is the same as counting back one from the previous difference.
4-3 – 4 – 2’ = ‘(4-2) = ‘2 = 1
If we used B for counting back by one
B(x) = ‘x
We can then define the B(x,y) function by
B(x,0) = x
B(x,y’) = ‘B(x,y)
where y must be less than or equal to x. Alternatively,
we could say that B has to be non-zero to continue. If we start y at 0, the stopping condition can be stated as stop when y equals x. So we don’t have to have less than or equal worked out yet.
This is the function given above for subtraction using the minus sign.
The B symbol makes clear that the structure of the B function is recursive.
If we continued to go backward when B reaches zero, then we would need to create negative numbers. This requires defining negative numbers as an ordered pair of a sign and size, and then redefining B on such ordered pairs. B is then a new function on these new types of inputs.