4 research outputs found
Towards computable analysis on the generalised real line
In this paper we use infinitary Turing machines with tapes of length
and which run for time as presented, e.g., by Koepke \& Seyfferth, to
generalise the notion of type two computability to , where
is an uncountable cardinal with . Then we start the
study of the computational properties of , a real closed
field extension of of cardinality , defined by the
first author using surreal numbers and proposed as the candidate for
generalising real analysis. In particular we introduce representations of
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
Realisability for Infinitary Intuitionistic Set Theory
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