basic constructions:
strong axioms
further
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
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
There are many variations of the paradox, depending for instance on what precise definition of “well-ordered” (and “ordinal number”) one chooses.
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.
English translations of Burali-Forti’s 1897 contributions can be found in
Another early reference on the set-theoretic paradoxes is
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)