nLab Burali-Forti's paradox

Burali-Fortis 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


Burali-Forti's paradox


Burali-Forti’s paradox is a paradox of naive material set theory that was first observed by Cesare Burali-Forti. A closely related paradox that uses well-founded sets? instead of ordinals is sometimes called Mirimanoff’s paradox.

However, the paradox is not specific to material set theory and can be formulated in structural set theory or in type theory. When formulated in type theory, it is often called Girard’s paradox after Jean-Yves Girard (see at type of types).

The paradox

Suppose that there were a set OrdOrd of all ordinal numbers. One could then prove that

  1. The set OrdOrd is well-ordered by the relation <\lt on ordinals.
  2. Thus, its order type?, call it say Ω\Omega, is itself an ordinal number.
  3. Thus Ω\Omega is an element of OrdOrd, which implies Ω<Ω\Omega\lt\Omega.
  4. But this is provably impossible for any ordinal number.

There are many variations of the paradox, depending for instance on what precise definition of “well-ordered” (and “ordinal number”) one chooses.

In type theory: Girard’s paradox

As formulated in type theory by Jean-Yves Girard, the Burali-Forti paradox shows that the original version of Per Martin-Löf‘s type theory, which allowed a type of types TypeType containing itself as a term Type:TypeType \colon Type, is inconsistent, in the precise sense that it contains (non-normalizing) proofs of false. As a result, a type of all types is similarly inconsistent because a type of all types would contain itself.

Moreover, by an adaptation of the proof, one can construct a looping combinator in this type theory, which implies the undecidability of type-checking.

Set-theoretic Girard’s paradox

A universal family of sets is a family of sets consisting of a set UU with index set II and a function E:UIE:U \to I, such that for all sets AA, there is a unique element i AIi_A \in I and a bijection δ A:AE *(i A)\delta_A:A \cong E^*(i_A) from AA to the fiber of EE at i Ai_A.

The set-theoretic Girard’s paradox states that having a universal family of sets is inconsistent.


English translations of Burali-Forti’s 1897 contributions can be found in

  • J. van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic 1879-1931 , Harvard UP 1967.

Another early reference on the set-theoretic paradoxes is

  • D. Mirimanoff, Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles , L’enseignement Mathématique 19 (1917) pp.37-52. (pdf; 19,4MB)

Girard’s paradox is discussed in

  • Per Martin-Löf, section 1.9, p. 7 of An intuitionistic theory of types: predicative part, In Logic Colloquium (1973), ed. H. E. Rose and J. C. Shepherdson (North-Holland, 1974), 73-118. (web)

  • Thierry Coquand, An analysis of Girard’s paradox , pp.227-236 in Proc. 1st Ann. Symb. Logic in Computer Science , IEEE Washington 1986. (ps-draft)

category: paradox

Last revised on November 24, 2023 at 14:41:04. See the history of this page for a list of all contributions to it.