Burali-Forti's paradox



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-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
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, (idemponent) 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. 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 containg itself as a term Type:TypeType \colon Type, is inconsistent, in the precise sense that it contains (non-normalizing) proofs of false.

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


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)

category: paradox

Revised on May 1, 2015 20:39:05 by Urs Schreiber (