Cantor developed the terms cardinal and ordinal number to teach arithmetic of infinite numbers. Dedekind’s book is about the natural numbers. However, Cantor’s focus on infinite number arithmetic is what is the focus of axiomatic set theory books. Because of this, these books are unreadable to most pre-service or in service teachers of K-3.
Dedekind’s focus on the natural numbers by themselves was lost to math because they were focused on the infinite numbers and associated problems with those.
When 1960s New Math came along, they were using math materials that were focused on abstract set theory or like the Landau book on Peano Axioms, had dropped most of the insight and teaching of the Dedekind book.
Thus 1960s New Math introduced bits and pieces of Dedekind and Cantor but there was no unifying text of math that was oriented towards what was needed for teaching K-3 or preservice teachers or standard setters or vendors. Thus there was confusion. This is why New Math failed in part. It is also a problem for common core standards today. That also has pieces of new math in it, such as 1 to 1 correspondences, but it fails to tie it together and relate defining addition by cardinality of the union of disjoint sets to counting on for example.
The subject of how definitions of addition link to order and how a chain of counting on numbers simultaneously defines addition and less than is not an insight articulated or made a goal to teach to students or preservice teachers. This type of lack of connection and structure is all through the materials for K-3, the standards, and the materials for preservice teachers.