|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|logical conjunction||product||product type|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator, (idemponent) monad||modal type theory, monad (in computer science)|
|linear logic||(symmetric, closed) monoidal category||linear type theory/quantum computation|
|proof net||string diagram||quantum circuit|
|(absence of) contraction rule||(absence of) diagonal||no-cloning theorem|
|synthetic mathematics||domain specific embedded programming language|
A first-order theory is a theory written in the language of first-order logic i.e it is a set of formulas or sequents (or generally, axioms over a signature) whose quantifiers and variables range over individuals of the underlying domain, but not over subsets of individuals nor over functions or relations of individuals etc. A first-order theory is called infinitary when the expressions contain infinite disjunctions or conjunctions, else it is called finitary.
Another name for a first-order theory is elementary theory which is found in the older literature but by now has gone extinct though it has left its traces in names like elementary topos, elementary embedding, ETCC etc.
Formulations of set theory are usually first-order theories, such as
In particular, the precise formulation of the former in first-order predicate logic by Skolem in 1922 provided the impetus for the hold that first-order predicate logic gained over mathematical logic in the following.
Let be a finitary first-order theory. There exists a Grothendieck topos and a -model in such that the first-order sequents satisfied in are precisely those provable in .
If is a theory over a countable signature, then can be taken as the topos of sheaves on the Cantor space . This completeness result shows that plays in constructive first-order logic a role similar to the role of in classical first-order logic.
Carsten Butz, Peter Johnstone, Classifying toposes for first-order theories , APAL 91 (1998) pp.33-58.
Peter Johnstone, Sketches of an Elephant II , Oxford UP 2002. (D1.3, D3.1, pp.899f)