Towards Computable Analysis on the Generalised Real Line

Abstract

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.

The Theory of the Generalised Real Numbers and Other Topics in Logic

“…We show that, while for countable families the only issues arising are of K-theoretic nature, for larger families the size itself becomes an obstacle. Using a combinatorial argument which goes back to Luzin's families, we prove, for a fairly general class of separable, nonabelian, nonunital C * -algebras A, the existence of a set of ℵ 1 commuting elements in Q(A) containing no uncountable subset that lifts to a set of commuting elements in M(A) (see also [6]). [5] , Voiculescu's theorem for nonseparable C * -algebras, arXiv preprint, 2018, arXiv:1811.09352.…”

“…[5] , Voiculescu's theorem for nonseparable C * -algebras, arXiv preprint, 2018, arXiv:1811.09352. [6] , Obstructions to lifting abelian subalgebras of corona algebras. Pacific Journal of Mathematics, vol.…”

“…Finally, we use the generalised version of type two Turing machines to begin the development of a generalised version of the classical theory of Weihrauch degrees (cf. [6]).…”

Realisability for Infinitary Intuitionistic Set Theory

“…To allow OTMs to work on sets we will follow the approach of computable analysis and use the coding that was introduced in Section 2.2 to define a notion of computability over sets (see also [10,1]).…”

Cheap Non-Standard Analysis and Computability: Some Applications

“…There must exist some cheap non-standard computable rationals p q and p q such that x − p q ≤ and y − p q ≤ . Then 5 Formally, we are using implicitely Lemma 44, which is easy to establish from definitions. bounding the in term K + by 1.…”