foundations

## Summary

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).

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 $Type$ containing 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.

A universal family of sets is a family of sets consisting of a set $U$ with index set $I$ and a function $E:U \to I$, such that for all sets $A$, there is a unique element $i_A \in I$ and a bijection $\delta_A:A \cong E^*(i_A)$ from $A$ to the fiber of $E$ at $i_A$.

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)