Dedekind wants to consider a transform function f(x), or x’ so that it is one to one from the domain to its range. This will end up being the successor function on natural numbers.
The closed sets under successor are tails of the naturals. So start from 2 and go on forever is the 2 tail. When we take the successor of the 2 tail we get the 3 tail. The 3 tail is also closed. So we have a 1 to 1 function from a closed set to a closed set.
This makes it very easy to work through the small lemmas in Dedekind’s book one after another. Here we refer to his 1888 book on the natural numbers, “What are numbers and what should they be?”
When we try to use initial segments or head segments from 0 to n or use chains from i to j, we have to deal with the extra complexity. If we use the predecessor function, and we use head segments from 0 to n, then ‘0 will not be defined or we have to start giving it a definition. Either it is 0, which destroys 1 to 1, or we have to start making up extended structures beyond zero. We could call those (-,i). These let us keep the predecessor as 1 to 1. But we now have these extra objects to deal with.
The path Dedekind ended up choosing or constructing avoids these complications.