Abstract
One objective of this paper is the determination of the proof-theoretic strength of Martin-Löf's type theory with a universe and the type of well-founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with Δ 12 comprehension and bar induction. As Martin-Löf intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the proof-theoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts.
Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman.
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The second author would like to thank the National Science Foundation of the USA for support by grant DMS-9203443
This article was processed by the author using the LATEX style filepljour1 from Spinger-Verlag.
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Griffor, E., Rathjen, M. The strength of some Martin-Löf type theories. Arch Math Logic 33, 347–385 (1994). https://doi.org/10.1007/BF01278464
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DOI: https://doi.org/10.1007/BF01278464