THEORIES
First-order and higher-order theories: systems individuated by a non-logical signature and the axioms governing it, with the consequence relation inherited from a base logic rather than defined. Peano arithmetic does not have a logic of its own; it has classical first-order logic plus 0, S, +, ×, and induction. What makes it an object is the signature and the axioms, and what makes it worth an entry is that most of the collection's metatheory is about theories of this kind.
The division is organized by what is axiomatized. Arithmetic entries axiomatize number and its fragments. Set entries axiomatize membership, classical and constructive. Truth entries axiomatize a truth predicate over a base theory. Part entries axiomatize parthood. Spacetime entries axiomatize physical structure — one entry, because the first-order axiomatization of a physical theory has been carried through seriously once, for relativity, and a group with one member is what the field has. These five groups are held flat in the division; the entries name their own subject and no further sorting is needed.
The Structure subdivision is the sixth group and the only one held separately, because it is the only one whose membership is not evident from the entry's name. Dense linear orders, well-orderings, algebraically closed and real closed fields, Tarski's elementary geometry, Boolean algebras, and the theories of groups and rings are theories in exactly the sense the criteria state, but they look like model theory, and their results are stated in Metatheory/Model-Theory with these theories as the instances. They have their own criteria file for that reason. The five flat groups axiomatize what mathematics is built from — or, in the spacetime case, what it is applied to — and are measured by their strength. Structure axiomatizes what mathematics is built into and is measured by its models: by what it decides.
Reading Structure against the arithmetic group is why it is here rather than in Applications. Peano arithmetic is incomplete and real closed fields are complete and decidable, in the same logic, with the same kind of axioms, over a domain that contains the naturals. The difference is definability: RCF cannot define ℕ, so nothing in it codes sequences, so Gödel's argument has no purchase. Presburger and Skolem arithmetic make the same point from the arithmetic side by removing an operation. The decidable theories are not a separate subject from the incomplete ones; they are the control.
The five apparatus divisions above—Algebraic, Modal, Structural, Type, Team—individuate by how a consequence relation is defined. This division individuates by what is asserted once a consequence relation is in hand. That is a different cut, and it is declared here for the same reason Applications declares its own: a reader who expects an apparatus axis and finds a signature axis should be told, not left to infer.