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
logic | category theory | type theory |
---|---|---|
true | terminal object/(-2)-truncated object | h-level 0-type/unit 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
logical 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
</table>
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)
Last revised on October 28, 2016 at 06:48:15. See the history of this page for a list of all contributions to it.