Richard Dedekind in 1888 did order before addition. Dedekind first did properties of closed sets under a function that is 1 to 1. Then he applied this to tail sets under successor in effect. Tail sets are natural numbers from some number on. So 2 to infinity including 2 is the 2 tail set. Then Dedekind did the initial segments and then he did the Recursion Theorem and then he did addition. Initial segments for Dedekind are from 1 to n and include 1 and n.
Subsequent authors try to avoid doing order first because it requires a long series of small lemmas. Those lemmas as Dedekind constructed them and proved them are actually the real meaning of order of natural numbers. Doing addition first does not help it hinders.
It would be nice to do the initial segments or head segments first. For New Math Done Right, head segments start from 0.
Professor David Groisser has started with head segments. This is tricky and he has a very clever approach to it.
For Professor Groisser the natural numbers start with 1.
Internet searches revealed no recognition of the importance of Professor Groisser’s work. So if one is taking a MOOC from a top tier school on this subject, you are actually missing out on the innovative work. In fact, innovative work in this area such as by Profs Groisser, Leonard Blackburn, David Pierce and others is generally not at the schools doing MOOCs.
So if one thinks that a MOOC from a top tier school is superior to a course given by an interested professor at another school, that is wrong. The innovations in this area are coming from below top tier schools and are ignored by the top tier schools including their MOOCs.
A gapless set can be defined as a set of natural numbers that contains the predecessor of each member of the set except for one member and contains the successor of each member of the set except for up to one member. If the number is zero, it is a tail. If it is one, it is a finite gapless set. Professor Groisser finds a way to define gapless head segments and prove their properties. This is tricky work. This is discussed further in Pre-Algebra New Math Done Right Peano Axioms.