This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype… Expand

The determination of the proof-theoretic strength of Martin-Löf's type theory with a universe and the type of well-founded trees is determined, showing that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic.Expand

It is verified (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in G coincides with theCollection of those functions which are Σ 1 -definable in KP − + Σ1 -Foundation + ∀ x ∃! y φ( x, y ).Expand

It is shown that a certain ordinal notation system is sufficient to measure the proof-theoretic strength of KPM, which involves a detour through an infinitary calculus RS(M), for which several cutelimination theorems are proved.Expand

A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their… Expand

It is shown that Kleene realizability provides a self-validating semantics for CZF, and this semantics is put to use in establishing several equiconsistency results, including theories of equal proof-theoretic strength with the same stock of provably recursive functions.Expand

A variant of realizability for Heyting arithmetic which validates Church’s thesis with uniqueness condition from its general form in intuitionistic set theory, IZF is introduced and several interesting corollaries are obtained.Expand