Dedekind did his book in 1888 using tail sets. He did order before recursion theorem and then addition. Dedekind uses intersection arguments early on in his book before order and even before tail sets.
Peano put addition in the axioms in 1889. He calls it a definition and then he argues that it is sufficient just to define addition recursively. But Dedekind and subsequent authors to Peano have rejected the Peano argument and gone back to either treating this definition as an axiom or proving a recursion theorem. Either way, Peano muddied the waters on the need for a recursion theorem and how to introduce addition.
Kalmar created a complicated definition of addition that people are still not sure if it is valid before the recursion theorem.
Complicated recursion theorems have been developed that can come before order. Then they define addition and define order from that.
David Groisser developed a definition of initial segments. He also uses an intersection argument at one point.
When we try to work with a chain from i to j right after the Peano Axioms, we find it is difficult. We don’t know which is first. We have not yet defined that. Our intersection definition may run into trouble. We can get into trouble because we assume we know if i is before j when we don’t have the right to.
We have to come up with conditions that lead us to the right answer. Some seemingly obvious combinations of conditions fail to work.
This is really what it means to understand order of natural numbers. When you go through all this difficulty, you realize that the natural numbers are a way to introduce non-local order, i.e. to compare two numbers that are not a predecessor successor pair.
Addition before order hides all this complexity. So people don’t understand where order comes from. That is a big gap in knowledge. This means professors, teachers and students all don’t understand the order of natural numbers. That translates into a total fail to understand the concept of numbers. This is part of why elementary education reverts to math fact democracy. Every idea is presented as equally important and logical with no order. This creates the feeling of randomness in math that people complain of. One random exercise after another with no structure, plan or meaning.
To have meaning, there must be a story. The story is absent when we take away the progression from Peano Axioms to fractions. What is left is random exercises and drills. This is disorienting and meaningless. It is all works and no vision. This fails to satisfy. So people get bored. They look for the answer in more works, i.e. more applications and fail to see it is the vision that is short changed not the applications.