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Computability on the Countable Ordinals and the Hausdorff-Kuratowski Theorem (Extended Abstract)

  • Arno PaulyEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9234)

Abstract

While there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far. In order to remedy this, we explore various potential representations of the set of countable ordinals. An equivalence class of representations is then suggested as a standard, as it offers the desired closure properties. With a decent notion of computability on the space of countable ordinals in place, we can then state and prove a computable uniform version of the Hausdorff-Kuratowski theorem.

Notes

Acknowledgements

I am grateful to Victor Selivanov for sparking my interest in a computable version of the Hausdorff Kuratowski theorem and to Vasco Brattka and Matthew de Brecht for various discussions on this question. The comparison of the various representations of the countable ordinals started with a discussion with Vassilios Gregoriades.

This work benefited from the Royal Society International Exchange Grant IE111233 and the Marie Curie International Research Staff Exchange Scheme Computable Analysis, PIRSES-GA-2011- 294962.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Clare CollegeUniversity of CambridgeCambridgeUK

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