P74 Remark 3.3.1. (i) One well-known consequence of the axiom of foundation of axiomatic set theory is the assertion that “∈-loops” a ∈ b ∈ c ∈ ... ∈ a can never occur in the set theory in which one works. On the other hand, there are many situations in mathematics in which one wishes to somehow “identify” mathematical objects that arise at higher levels of the ∈-structure of the set theory under consideration with mathematical objects that arise at lower levels of this ∈-structure.
P75 That is to say, the mathematical objects at both higher and lower levels of the ∈-structure constitute examples of the same mathematical notion of a “set”, so that one may consider “bijections of sets” between those sets without violating the axiom of foundation. In some sense, the notion of a species may be thought of as a natural extension of this observation. That is to say, the notion of a “species” allows one to consider, for instance, speciesisomorphisms between species-objects that occur at different levels of the ∈-structure of the set theory under consideration - i.e., roughly speaking, to “simulate ∈-loops” - without violating the axiom of foundation.
