nLab Curry's paradox





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




In mathematical logic and mathematical foundations Curry’s paradox is a paradox which is a version of Russell's paradox that does not involve the use of negation.

Russell’s paradox is a proof of falsehood, rendering a certain class of formal systems (those including some kind of unlimited axiom of comprehension) “inconsistent” in the sense that they prove false, which by the law of ex falso quodlibet entails that they prove anything at all. Curry’s paradox is a direct proof of “anything at all” without the detour through falsehood. Thus, it applies more generally than Russell’s paradox, e.g. to theories lacking a notion of “false”, or to paraconsistent theories whose notion of “false” does not satisfy ex falso quodlibet.

Curry’s paradox does, however, still depend on the structural rule of contraction. Thus, it can be avoided by systems based on linear logic.


Let PP be any statement at all, and consider the set

C={x(xx)P}. C = \{ x \mid (x\in x) \Rightarrow P \}.

Then if CCC\in C, by definition we have (CC)P(C\in C) \Rightarrow P, and hence by modus ponens we have PP. Therefore, (CC)P(C\in C) \Rightarrow P. But by definition this means that CCC\in C, and therefore (as we just proved) PP.

Note that, if negation is defined as ¬P=(P)\neg P = (P \Rightarrow \bot) for some notion of falsehood \bot, then Russell’s paradox is the special case of Curry’s paradox for P=P=\bot.


In relation to the axiom of full comprehension:

Last revised on September 18, 2021 at 05:12:20. See the history of this page for a list of all contributions to it.