 At this point, we can go down one of two “abstractionist” paths. First, we can simply identify the relevant structure with that equivalence class (with the “concept in extension”, as it is sometimes put). Thus, corresponding to the concept “natural number system” and set-theoretic systems falling under it, like the finite von Neumann ordinals, there is the entire equivalence class of corresponding models as a third entity. (Our primary focus is again on categorical axiom systems, but the approach can be generalized.) It is this class that is now called “the natural number structure”. It is not a set, to be sure, but a proper class; but it can still be studied logico-mathematically.  Looked at from a “relativist” perspective, the new approach can also be described as follows: Its core is to go from a particular, arbitrarily chosen system falling under a higher-order concept to the related equivalence class, i.e., the class of all systems isomorphic to it (in the categorical case).  And we can think of this move as involving a kind of “abstraction”, specifically in the sense of the “principle of abstraction” in Russell (1903; also adopted by Rudolf Carnap and others). The result is a first form of “abstractionist structuralism”.