Realisability for Infinitary Intuitionistic Set Theory

Abstract

We introduce a realisability semantics for infinitary intuitionistic set theory that employs Ordinal Turing Machines (OTMs) as realisers. We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary intuitionistic logic, and that all axioms of infinitary Kripke-Platek set theory are realised. As an application of our technique, we show that the propositional admissible rules of (finitary) intuitionistic Kripke-Platek set theory are exactly the admissible rules of intuitionistic propositional logic.

The First-Order Logic of CZF Is Intuitionistic First-Order Logic

“…Visser [25] proved that the propositional admissible rules of Heyting Arithmetic HA are exactly the admissible rules of intuitionistic propositional logic IPC. Using realisability techniques, Carl, Galeotti, and Passmann [4] determined the propositional admissible rules of IKP to be exactly the admissible rules of propositional intuitionistic logic. Iemhoff and Passmann [12] proved that the propositional admissible rules of CZF ER and IZF R are the admissible rules of intuitionistic propositional logic by using a modification of the so-called blended models (earlier introduced by Passmann [20]).…”

“…The other direction can be shown by a straightforward but tedious coding argument by using a large enough fragment of the well-order < as a parameter (Carl, Galeotti, and Passmann [4] spell out a very similar argument in an appendix; Carl [3, Section 2.3.2 and Chapter 3] discusses codings as well).…”

“…Similarly, we say that ϕ is SRM + -realisable if and only if there is an SRM + realising ϕ; and so for SRM +,O , SRM + L , and SRM +,O L . This could be extended to infinitary languages as done by Carl, Galeotti, and Passmann [4]. Analogously to (i) and (ii), one could give realisability semantics to the global well-order < .…”

“…Finally, we can use SRM + -realisability to easily determine the admissible rules of CZF. A proof of Carl, Galeotti, and Passmann [4,Theorem 56] can be adapted to work here. Proof.…”

“…Carl, Galeotti, and Passmann [4] gave a first proof-theoretic application of transfinite computability and provided a realisability interpretation for (infinitary) IKP set theory using OTMs. In particular, they proved that the propositional admissible rules of IKP are exactly the admissible rules of intuitionistic propositional logic.…”