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Towards Computable Analysis on the Generalised Real Line

  • Lorenzo Galeotti
  • Hugo Nobrega
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10307)

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 \({{\mathrm{\mathbb {R}}}}_\kappa \), a real closed field extension of \({{\mathrm{\mathbb {R}}}}\) 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 \({{\mathrm{\mathbb {R}}}}_\kappa \) 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.

Keywords

Turing Machine Baire Space Cantor Space Uncountable Cardinal Real Closed Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This research was partially done whilst the authors were visiting fellows at the Isaac Newton Institute for Mathematical Sciences in the programme Mathematical, Foundational and Computational Aspects of the Higher Infinite. The research benefited from the Royal Society International Exchange Grant Infinite games in logic and Weihrauch degrees. The second author was also supported by the Capes Science Without Borders grant number 9625/13-5. The authors are grateful to Benedikt Löwe and Arno Pauly for the many fruitful discussions and to the Institute for Logic, Language and Computation for the hospitality offered to the first author. Finally, the authors wish to thank the three anonymous referees for the helpful comments which have improved the paper.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität HamburgHamburgGermany
  2. 2.Institute for Logic, Language and ComputationUniversiteit van AmsterdamAmsterdamThe Netherlands

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