nLab
first-order theory

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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-level 1-type/h-prop
proofgeneralized elementprogram
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 monadmodal type theory, monad (in computer science)

homotopy levels

semantics

Contents

Idea

A first-order theory is a theory which allows for quantification over variables, but not over sets of variables. It is a theory (= language + axioms) in a language consisting of the language of first-order logic, plus additional relation, function and constant symbols.

Examples

Formulation of set theory is a first-order theory, such as

Revised on November 20, 2012 12:07:02 by Urs Schreiber (82.169.65.155)
Edit | Back in time (1 revision) | See changes | History | Views: Print | TeX | Source | Linked from: allegory, Elephant, completion, theory, syntactic category, pretopos completion, elementary class of structures, categoricity, elementary function arithmetic