## Counting back and predecessor function

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.