Define prime by 1=0′, 2 = 1′, 3 = 2′, 4=3′, etc.
The counting on identities are
2+0 = 2
2+3′ = (2+3)’
Here we call 2 the anchor or base and 3 the shift. The latter is not standard, anchor is standard in counting on lessons.
Anchor has the advantage over base of not being confused with base 10.
The cardinality rule for adding two numbers is actually more complicated in practice.
You have 2 and want to add 4. You find two sets with no elements in common. One is in bijection (1 to 1 correspondence) with the segment from 1 to 2 inclusive, the counting 2 head segment and the other from 1 to 4 inclusive.
Or you can use the head segment 0 to 1 and combine with 0 to 3. But you still have to distinguish them.
This can be done by forming ordered pairs.
(A,0), (A,1) is the head segment for set A
for set B.
Notice we have to introduce ordered pairs.
Now we form the union of these two sets.
So we have the set
Now we still have to count the elements.
Counting on is a way to start from the end of the
A segment and count on by the B segment.
The counting on identities are formulas. We want addition as a formula, 2 + 3. Defining the addition formula in terms of the counting on function 2′, 3′, n’, is easier than taking two head segments, forming ordered pairs within each to distinguish the two head segments, combining these variant head segments into a single set and then counting from scratch the combined set.
So we see that the idea of combining two sets of distinct elements as a way to define addition sounds good, and is easy to prove theorems like commutative and associative laws of addition, but it ends up being more complicated in terms of how you actually implement it.
If you want to think of addition as a formula, then it is easier to define it as a formula in terms of counting on as a formula.
x+0 = x
x+y’ = (x+y)’
This defines addition, x+y, as a formula in terms of a formula for succession, the prime symbol, y’, and (x+y)’.
This is why it is better to base addition on counting on instead of cardinality of the union of two sets. The latter requires finding two sets that don’t overlap.
If you start with head segments, they already overlap, so you have to distinguish them somehow, i.e. find other sets in bijection with them. Or at least one such auxiliary set. Then you have to mechanically count on from scratch anyhow in the combined set.
So you might as well count on from the end of the base segment using the shift segment and using the counting on identities.
If you count on from scratch the combined set, you will end up retracing the counting on from the end of the anchor (or base) head segment anyhow.
The importance of this discussion is because cardinality is what was used under 1960s New Math and is still used today. However, as we see, cardinality is actually not as good as the counting on identities for defining addition.
Cardinality in practice is 1) cumbersome 2) lacks an explicit formula 3) you end up with counting on anyhow.
So you might as well work with the counting on identities to define addition.
Whichever way one starts, one should prove the other one follows. But that will be a topic for another post or left in the book series forthcoming on this topic.