The real line R and Baire space are two of the most fundamental objects in descriptive set theory and have been studied extensively. In recent years, set theorists have become increasingly interested in generalised Baire spaces κ κ , i.e., the sets of functions from κ to κ for an uncountable cardinal κ. In the thesis, two κ-analogues of R are discussed, one at https://www.cambridge.org/core/terms.

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Abstract. In this paper we use infinitary Turing machines with tapes of length κ and which run for time κ as presented, e.g., by Koepke & Seyfferth, to generalise the notion of type two computability to 2 κ , where κ is an uncountable cardinal with κ <κ = κ. Then we start the study of the computational properties of Rκ, a real closed field extension of R of cardinality 2 κ , defined by the first author using surreal numbers and proposed as the candidate for generalising real analysis. In particular we introduce representations of Rκ under which the field operations are computable. Finally we show that this framework is suitable for generalising the classical Weihrauch hierarchy. In particular we start the study of the computational strength of the generalised version of the Intermediate Value Theorem.

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.

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