Skip to main content Accessibility help
×
Home

Computably Isometric Spaces

  • Alexander G. Melnikov (a1)

Abstract

We say that an uncountable metric space is computably categorical if every two computable structures on this space are equivalent up to a computable isometry. We show that Cantor space, the Urysohn space, and every separable Hilbert space are computably categorical, but the space [0, 1] of continuous functions on the unit interval with the supremum metric is not. We also characterize computably categorical subspaces of ℝ n , and give a sufficient condition for a space to be computably categorical. Our interest is motivated by classical and recent results in computable (countable) model theory and computable analysis.

Copyright

References

Hide All
[1] Ash, C. and Knight, J., Computable structures and the hyperarithmetical hyerarchy, Elsevier, Amsterdam, 2000.
[2] Bienvenu, L., Holzl, R, Miller, J., and Nies, A., The denjoy alternative for computable functions, Symposium on theoretical aspects of computer science, STACS 2012 (Dürr, Christoph and Wilke, Thomas, editors), LIPIcs, vol. 14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012, pp. 543554.
[3] Brattka, V., Hertling, P., and Weihrauch, K., A tutorial on computable analysis, New computational paradigms: Changing conceptions of what is computable (Cooper, S. Barry, Löwe, Benedikt, and Sorbi, Andrea, editors), Springer, 2008, pp. 425491.
[4] Brouwer, L., Collected works. Vol. 1, North-Holland, Amsterdam, 1975, Philosophy and foundations of mathematics, Edited by Heyting, A..
[5] Brouwer, L., Collected works, Vol. 2, North-Holland, Amsterdam, 1976, Geometry, analysis, topology and mechanics, Edited by Freudenthal, Hans.
[6] Downey, R., Computability theory and linear orderings, Handbook of recursive mathematics, Vol. 2, Studies in Logic and the Foundation of Mathematics, vol. 139, North-Holland, Amsterdam, 1998, pp. 823976.
[7] Downey, R., Kach, A., Lempp, S., and Turetsky, D., On computable categoricity, To appear.
[8] Downey, R. and Hirschfeldt, D., Algorithmic randomness and complexity, Theory and Applications of Computability, Springer, 2010.
[9] Ershov, Yu and Goncharov, S., Constructive models, Kluwer Academic Publications, 2000.
[10] Ershov, Yuri L., Theory of numberings, Handbook of computability theory, Studies in Logic and the Foundations of Mathematics, vol. 140, North-Holland, Amsterdam, 1999, pp. 473503.
[11] Fröhlich, A. and Shepherdson, J., Effective procedures in field theory, Philosophical Transactions of the Royal Society of London. Series A., vol. 248 (1956), pp. 407432.
[12] Goncharov, S., Autostability of models and abelian groups, Algebra and Logic, vol. 19 (1980), pp. 1327, (English translation).
[13] Goncharov, S., The problem of the number of nonautoequivalent constructivizations, Algebra and Logic, vol. 19 (1980), pp. 1327, (English translation).
[14] Goncharov, S., Countable boolean algebras and decidability, Siberian School of Algebra and Logic, Nauchnaya Kniga, Novosibirsk, 1996.
[15] Hertling, P., A real number structure that is effectively categorical, Mathematical Logic Quarterly, vol. 45 (1999), no. 2, pp. 147182.
[16] Iljazović, Z., Isometries and computability structures, Journal of Universal Computer Science, vol. 16 (2010), no. 18, pp. 25692596.
[17] Katětov, M., On universal metric spaces, General topology and its relations to modern analysis and algebra, VI (Frolik, Z., editor), vol. 16, Heldermann, Berlin, 1988, pp. 323330.
[18] Kushner, B., Lectures on constructive mathematical analysis, Translations of Mathematical Monographs, vol. 60, American Mathematical Society, Providence, Rhode Island, 1984.
[19] LaRoche, P., Recursively presented boolean algebras, Notices AMS, vol. 24 (1977), pp. 552553.
[20] Mal′Cev, A., On recursive Abelian groups, Doklady Akademii Nauk SSSR, vol. 32 (1962), pp. 14311434.
[21] Melleray, J., Some geometric and dynamical properties of the Urysohn space, Topology and its Applications, vol. 155 (2008), no. 14, pp. 15311560.
[22] Melnikov, A. and Ng, K.M., Computable structures and operations on the space of continuous functions, to appear.
[23] Metakides, G. and Nerode, A., Effective content of field theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289320.
[24] Miller, R. and Schoutens, H., Computably categoricalfields viafermat's last theorem, to appear.
[25] Myhill, J., A recursive function, defined on a compact interval and having a continuous derivative that is not recursive, The Michigan Mathematical Journal, vol. 18 (1971), pp. 9798.
[26] Nies, A., Computability and randomness, Oxford Logic Guides, vol. 51, Oxford University Press, Oxford, 2009.
[27] Nies, A., Interactions of computability and randomness, Proceedings of the International Congress of Mathematicians. Volume II (New Delhi) (Ragunathan, S., editor), Hindustan Book Agency, 2010, pp. 3057.
[28] Nurtazin, A.T., Computable classes and algebraic criteria of autostability, Summary of Scientific Schools, Mathematical Institute SB of USSR AS, Novosibirsk, 1974.
[29] Pour-El, M. and Richards, I., Computability and noncomputability in classical analysis, Transactions of the American Mathematical Society, vol. 275 (1983), no. 2, pp. 539560.
[30] Pour-El, M. and Richards, J., Computability in analysis and physics, Perspectives in Mathematical Logic, Volume 1, Springer, 1989.
[31] Rabin, M., Computable algebra, general theory and theory of computable fields., Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341360.
[32] Remmel, J. B., Recursively categorical linear orderings, Proceedings of the American Mathematical Society, vol. 83 (1981), pp. 387391.
[33] Turing, A., On Computable Numbers, with an Application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, vol. 42 (1936), no. 2, pp. 230265.
[34] Turing, A., On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction, Proceedings of the London Mathematical Society, vol. 43 (1937), no. 2, pp. 544546.
[35] Urysohn, P., Sur un espace métrique universel, Bulletin des Sciences Mathématiques, vol. 51 (1927), pp. 43–64, 7490.
[36] Uspenskij, V., The Urysohn universal metric space is homeomorphic to a Hilbert space, Topology and its Applications, vol. 139 (2004), pp. 145149.
[37] Weihrauch, K., Computable analysis, Springer, 2000.
[38] White, W., On the complexity of categoricity in computable structures, Mathematical Logic Quarterly, vol. 49 (2003), no. 6, pp. 603614.

Keywords

Related content

Powered by UNSILO

Computably Isometric Spaces

  • Alexander G. Melnikov (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.