nLab propositional axiom of choice





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
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




The propositional axiom of choice is simply another name for the usual axiom of choice in dependent type theory, to distinguish with the untruncated version of the axiom of choice called the type theoretic axiom of choice.

It corresponds to the internal axiom of choice in categorical semantics and in set theory.


The propositional 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 R:( x:Ay:B.R(x,y))(f:AB. x:AR(x,f(x)))\mathrm{ac}_R:\left(\prod_{x:A} \exists y:B.R(x, y)\right) \to \left(\exists f:A \to B.\prod_{x:A} R(x, f(x))\right)

The propositions as subsingletons interprets disjunction and existential quantification as the bracket type of the sum type and dependent sum type respectively, and leads to the usual statement of the axiom of choice translated to dependent type theory.

The equivalent form of the axiom of choice involving a unary type family x:AP(x)x:A \vdash P(x) instead of a binary type family x:A,y:BR(x,y)x:A, y:B \vdash R(x, y) states that one can construct

ac R:( x:A[P(x)])[ x:AP(x)]\mathrm{ac}_R:\left(\prod_{x:A} [P(x)]\right) \to \left[\prod_{x:A} P(x)\right]

This is the statement that any dependent product of any family of inhabited h-sets is inhabited.


  • Martin Hyland, p. 1 of: Variations on realizability: realizing the propositional axiom of choice, Mathematical Structures in Computer Science, Volume 12, Issue 3, June 2002, pp. 295 - 317 (doi:10.1017/S0960129502003651, pdf)

Last revised on February 5, 2024 at 01:50:41. See the history of this page for a list of all contributions to it.