If we define a head segment as 0 to n, and a tail as n to infinity, both including n, we see something interesting on intersections.
The tail starting at 2 is the intersection of the tail starting at 0, starting at 1 and starting at 2. Thus 2′ sets.
The head ending at 2 is the intersection of every
head that ends at 2 or afterwards, an infinite number.
If we go from 2 to 5, then we take the intersections of
all tails starting from 0, 1 or 2, and all heads that end at 5 or later.
At some point in every approach from Dedekind on, we have a point where we do one of these intersections.
Standard treatments do an intersection approach for the proof of the recursion theorem. This is usually complicated and difficult to understand.
Dedekind was the first to have some sort of intersection stage.
One can gain insight by determining what intersection of what sets is being used.