Towards computable analysis on the generalised real line

TL;DR

This paper extends the concept of computability to uncountable settings using infinitary Turing machines, introduces a new generalized real number system, and explores its implications for computability and analysis.

Contribution

It generalizes type two computability to uncountable cardinals, introduces the structure al R_, and adapts the Weihrauch hierarchy to this new framework.

Findings

Defined representations of al R_ with computable field operations.

Extended the Weihrauch hierarchy to the uncountable setting.

Initiated the study of the computational strength of the generalized Intermediate Value Theorem.

Abstract

In this paper we use infinitary Turing machines with tapes of length $\kappa$ and which run for time $\kappa$ as presented, e.g., by Koepke \& Seyfferth, to generalise the notion of type two computability to $2^{\kappa}$, where $\kappa$ is an uncountable cardinal with $\kappa^{<\kappa}=\kappa$. Then we start the study of the computational properties of $\mathbb{R}_\kappa$, a real closed field extension of $\mathbb{R}$ of cardinality $2^{\kappa}$, defined by the first author using surreal numbers and proposed as the candidate for generalising real analysis. In particular we introduce representations of $\mathbb{R}_\kappa$ 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.