As the previous post indicates, counting on is the basis of order. This is made most clear in the original Dedekind 1888 book “The nature and meaning of numbers.”
Dedekind uses the prime notation for successor, and this is the same as counting on by one.
One can do order the same way as Dedekind does in his book. This in effect is order from counting on.
This also shows that addition is linked to order and counting on by one. It is easy to think that addition is somehow a primitive operation independent of counting on by one and of order. This is incorrect. They are inextricably linked.
Addition is simply a subset of order and counting on by one. This is what the Peano Axioms show us. They are axioms for counting on by one. Even the axiom of induction is an axiom that counting on by one from zero will eventually reach every whole number.
The Dedekind book can be rewritten in terms of counting on by one instead of successor. I will add this to my project list.
The actual introduction of whole numbers and addition and counting on by one repeatedly to do addition is logically the same as the Grassmann Dedekind approach. What is different in the GD approach is that gaps in logic or assumptions are made explicit.
These actually help the pedagogy. When the lesson plan writer, text material writer, teacher, and parent are all unaware of the logical structure and the gaps in what is being said or how it is presented, the students can fall into those gaps and get stuck.
Knowing the material logically is how we identify why they are stuck and what explanation to use to get them unstuck. This requires a thorough learning of these materials by preservice and service teachers for grade K on up.