The inconsistency of higher order extensions of Martin-löf's type theory

Journal of Philosophical Logic 18 (4):399 - 422 (1989)
Martin-Löf's constructive type theory forms the basis of this paper. His central notions of category and set, and their relations with Russell's type theories, are discussed. It is shown that addition of an axiom - treating the category of propositions as a set and thereby enabling higher order quantification - leads to inconsistency. This theorem is a variant of Girard's paradox, which is a translation into type theory of Mirimanoff's paradox (concerning the set of all well-founded sets). The occurrence of the contradiction is explained in set theoretical terms. Crucial here is the way a proof-object of an existential proposition is understood. It is shown that also Russell's paradox can be translated into type theory. The type theory extended with the axiom mentioned above contains constructive higher order logic, but even if one only adds constructive second order logic to type theory the contradictions arise
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.1007/BF00262943
 Save to my reading list
Follow the author(s)
Edit this record
My bibliography
Export citation
Find it on Scholar
Mark as duplicate
Request removal from index
Revision history
Download options
Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 29,530
Through your library
References found in this work BETA

No references found.

Add more references

Citations of this work BETA

Add more citations

Similar books and articles
Added to PP index

Total downloads
22 ( #244,210 of 2,210,802 )

Recent downloads (6 months)
2 ( #207,996 of 2,210,802 )

How can I increase my downloads?

Monthly downloads
My notes
Sign in to use this feature