nLab first-order theory



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Model theory



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.

The characterization of set-theoretic models of (finitary) first-order theories is the topic of traditional model theory.


Formulations of set theory are usually first-order theories (see first-order set theory), 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.

It is somewhat ironic that classical first-order logic owes its promotion to prominence to intuitionistic mathematicians like Weyl (1910,1918) and Skolem.



Let 𝕋\mathbb{T} be a finitary first-order theory. There exists a Grothendieck topos \mathcal{E} and a 𝕋\mathbb{T}-model N 𝕋N_{\mathbb{T}} in \mathcal{E} such that the first-order sequents satisfied in N 𝕋N_{\mathbb{T}} are precisely those provable in 𝕋\mathbb{T}.

This result due to P. Freyd is discussed in Johnstone (2002, p.900) or Freyd-Scedrov (1990, p.130).

If 𝕋\mathbb{T} is a theory over a countable signature, then \mathcal{E} can be taken as the topos Sh(2 )Sh(2^{\mathbb{N}}) of sheaves on the Cantor space 2 2^{\mathbb{N}}. This completeness result shows that Sh(2 )Sh(2^{\mathbb{N}}) plays in constructive first-order logic a role similar to the role of SetSet in classical first-order logic.


Last revised on May 25, 2024 at 16:25:00. See the history of this page for a list of all contributions to it.