nLab type theoretic axiom of choice

Contents

Context

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms

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
propositionsetobjecttype
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

semantics

Contents

 Idea

In dependent type theory, the type theoretic axiom of choice is a version of axiom of choice which does not use any bracket types in the theorem, as it uses the propositions as types interpretation of dependent type theory. Despite its name, the type theoretic axiom of choice is provable in dependent type theory from the inference rules of dependent product types, dependent sum types, and identity types, and so really deserves to be called the type theoretic principle of choice. This is in contrast to what is referred to as the propositional axiom of choice or simply axiom of choice which does use bracket types via the propositions as subsingletons interpretation of dependent type theory and is just the usual axiom of choice expressed in dependent type theory.

Statement

The type theoretic axiom of choice is the statement that given types AA and BB and type family x:A,y:BR(x,y)x:A, y:B \vdash R(x, y), one can construct

ac Rλg.(λx.π 1(g(x)),λx.π 2(g(x))):( x:A y:BR(x,y))( f:AB x:AR(x,f(x)))\mathrm{ac}_R \coloneqq \lambda g.(\lambda x.\pi_1(g(x)), \lambda x.\pi_2(g(x))):\left(\prod_{x:A} \sum_{y:B} R(x, y)\right) \to \left(\sum_{f:A \to B} \prod_{x:A} R(x, f(x))\right)

The propositions as types interprets disjunction and existential quantification directly as the sum type and dependent sum type respectively, and the statement of the axiom of choice comes out as simply the statement that products distribute over coproducts. (See distributivity pullback for a discussion in terms of the internal type theory of a locally cartesian closed category.)

The equivalent form of the axiom of choice involving cartesian products of inhabited types then becomes in type theory the statement that “any dependent product of any family of pointed types is pointed”. and is just the identity function on the dependent product type.

 References

The type theoretic axiom of choice is found in:

Created on February 5, 2024 at 01:25:56. See the history of this page for a list of all contributions to it.