On Some Semi-constructive Theories Related to Kripke–Platek Set Theory

Part of the Outstanding Contributions to Logic book series (OCTR, volume 13)


We consider some very robust semi-constructive theories related to Kripke–Platek set theory, with and without the powerset operation. These theories include the law of excluded middle for bounded formulas, a form of Markov’s principle, the unrestricted collection scheme and, also, the classical contrapositive of the bounded collection scheme. We analyse these theories using forms of a functional interpretation which work in tandem with the constructible hierarchy (or the cumulative hierarchy, if the powerset operation is present). The main feature of these functional interpretations is to treat bounded quantifications as “computationally empty.” Our analysis is extended to a second-order setting enjoying some forms of class comprehension, including strict-\(\Pi ^1_1\) reflection. The key idea of the extended analysis is to treat second-order (class) quantifiers as bounded quantifiers and strict-\(\Pi ^1_1\) reflection as a form of collection. We will be able to extract some effective bounds from proofs in these systems in terms of the constructive tree ordinals up to the Bachmann–Howard ordinal.


Intuitionistic Kripke–Platek set theory Functional interpretations \(\Sigma \)-ordinal Strict-\(\Pi ^1_1\) reflection Power Kripke–Platek set theory 

MSC (2010)

03F10 03F35 03F65 


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Authors and Affiliations

  1. 1.Faculdade de Ciências, Departamento de MatemáticaUniversidade de LisboaLisboaPortugal

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