Summary

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).

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

1. The set $Ord$ is well-ordered by the relation $\lt$ on ordinals.
2. Thus, its order type?, call it say $\Omega$, is itself an ordinal number.
3. Thus $\Omega$ is an element of $Ord$, which implies $\Omega\lt\Omega$.
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 $Type$ containg itself as a term $Type \colon Type$, 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.

References

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)