INCLUSION CRITERIA
An entry belongs in this subdivision if and only if it is a theory whose signature axiomatizes an order, algebraic, or geometric structure, and whose interest is the class of its models rather than its foundational strength.
Required: Both of the following:
- A non-logical signature naming an order, operation, or geometric primitive, with axioms governing it, consequence inherited from classical first-order logic
- Individuation by the models: the entry is worth stating because a determinate class of structures satisfies it, and its metatheory (completeness, decidability, quantifier elimination, categoricity) is about that class
Not sufficient: Being a first-order theory, since every entry in this division is one. Being applied to a structure. A theory whose interest is what it can prove about itself or about other theories, rather than what satisfies it, belongs with the foundational groups above: arithmetic entries axiomatize number and its fragments, set entries membership, truth entries a truth predicate, part entries parthood. The test is what the entry is measured by — strength, or models.
Not sufficient: A model-theoretic property, technique, or theorem. Quantifier elimination, o-minimality, categoricity, and stability are properties that theories have; they belong in Metatheory/Model-Theory, and the theories that have them belong here. An entry stating a general result about a class of structures is Metatheory even if a particular structure motivates it.
Boundary: Arithmetic in a reduced signature is placed by what it axiomatizes, not by its behaviour: Presburger and Skolem arithmetic are decidable and admit quantifier elimination like the entries here, but they axiomatize number and stay with the arithmetic group. Tarski's elementary geometry is here and cross-listed to Applications/Spatial, where spatial reasoning systems live; the bare axiomatization is here, the reasoning apparatus is there. Theories of algebraic structures are here for their axiomatizations; the undecidability results proved by interpreting them are Metatheory, cross-listed from Robinson Arithmetic. An algebra used as the semantics of a logic rather than axiomatized as a structure — Boolean, Heyting, residuated, orthomodular — belongs in Algebraic, which individuates by the consequence relation the algebra defines.