nLab
first-order theory

Context

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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) 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

semantics

Model theory

Contents

Idea

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.

Examples

Formulations of set theory are usually first-order theories, such as

Remark

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.

Properties

Proposition

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.

References

Revised on November 22, 2016 05:08:52 by David Corfield (217.39.117.236)