Abstract
Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive definitions, subsystems of analysis and finite type systems. With the pioneering work of Gerhard Jäger in the late 1970 s and early 1980s, the focus switched to set theories, furnishing ordinal-theoretic proof theory with a uniform and elegant framework. More recently it was shown that these tools can even sometimes be adapted to the context of strong axioms such as the powerset axiom, where one does not attain complete cut elimination but can nevertheless extract witnessing information and characterize the strength of the theory in terms of provable heights of the cumulative hierarchy. Here this technology is applied to intuitionistic Kripke-Platek set theories \(\mathbf{IKP }({\mathcal {P}})\) and \(\mathbf{IKP }({\mathcal {E}})\), where the operation of powerset and exponentiation, respectively, is allowed as a primitive in the separation and collection schemata. In particular, \(\mathbf{IKP }({\mathcal {P}})\) proves the powerset axiom whereas \(\mathbf{IKP }({\mathcal {E}})\) proves the exponentiation axiom. The latter expresses that given any sets A and B, the collection of all functions from A to B is a set, too. While \(\mathbf{IKP }({\mathcal {P}})\) can be dealt with in a similar vein as its classical cousin, the treatment of \(\mathbf{IKP }({\mathcal {E}})\) posed considerable obstacles. One of them was that in the infinitary system the levels of terms become a moving target as they cannot be assigned a fixed level in the formal cumulative hierarchy solely based on their syntactic structure. As adumbrated in an earlier paper, the results of this paper are an important tool in showing that several intuitionistic set theories with the collection axiom possess the existence property, i.e., if they prove an existential theorem then a witness can be provably described in the theory, one example being intuitionistic Zermelo-Fraenkel set theory with bounded separation.
Dedicated to Gerhard Jäger on the occasion of his 60th birthday.
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Notes
- 1.
Although sometimes even set theories with strong choice principles can have the \({\mathbf {EP}}\) (see [21]).
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The work of the second author was supported by a Leverhulme Research Fellowship and the Engineering and Physical Sciences Research Council under grant number EP/K023128/1.
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Cook, J., Rathjen, M. (2016). Ordinal Analysis of Intuitionistic Power and Exponentiation Kripke Platek Set Theory. In: Kahle, R., Strahm, T., Studer, T. (eds) Advances in Proof Theory. Progress in Computer Science and Applied Logic, vol 28. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29198-7_4
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