Abstract
We discuss a relativization of real numbers to a universe given by a function algebra, and develop a tentative theory of relativized real numbers. We show that the class R(Ϝptime) of real numbers, obtained by relativizing to the class F Ptime of polynomial time computable functions, is a proper subclass of the class R(ε) of real numbers, obtained by relativizing to the class ε of elementary functions. We show the Cauchy completeness of relativized real numbers, and that we can prove the (constructive or approximate) intermediate value theorem if our universe is closed under a closure condition used to characterize the polynomial time computable functions.
Keywords
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Errett Bishop, Foundations of Constructive Mathematics, McGraw-Hill, New York, 1967.
Errett Bishop and Douglas Bridges, Constructive Analysis, Springer, Berlin, 1985.
Peter Clote, Computational models and functional algebras, In E.R. Griffor ed., Handbook of Computability Theory, North-Holland, Amsterdam, 1999.
Alan Cobham, The intrinsic computational difficulty of functions, In Y. Bar-Hillel ed., Logic, Methodology and Philosophy of Science II, North-Holland, Amsterdam, 24–30, 1965.
P. Csillag, Eine Bemerkung zur Auflosung der eingeschachtelten Rekursion, Acta Sci. Math. Szeged. 11, 169–173, 1947.
Hajime Ishihara, Function algebraic characterizations of the polytime functions, Comput. Complexity 8, 346–356, 1999.
Hajime Ishihara, Feasibly constructive analysis, Sūrikaisekikenkyūsho Kōkyūroku 1169, 76–83, 2000.
Hajime Ishihara, Constructive reverse mathematics: compactness properties, In L. Crosilla and P. Schuster eds., From Sets and Types to Analysis and Topology: Towards Practicable Foundations for Constructive Mathematics, Oxford University Press, Oxford, 245–267, 2005.
Neil D. Jones, Computability and Complexity: From a Programming Perspective, MIT Press, Cambridge, 1997.
László Kalmár, Egyszerü példa eldönthetetlen aritmetikai problémára, Mate és Fizikai Lapok. 50, 1–23, 1943.
Ker-I Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991.
Lars Kristiansen, Neat function algebraic characterizations of logspace and linspace, Comput. Complexity 14, 72–88, 2005.
Marian B. Pour-El and Jonathan I. Richards, Computability in Analysis and Physics, Springer, New York, Berlin, 1989.
Harvey E. Rose, Subrecursion: Functions and Hierarchies, Oxford University Press, Oxford, 1984.
Anne S. Troelstra and Dirk van Dalen, Constructivism in Mathematics, Vol. I. An Introduction, North-Holland, Amsterdam, 1988.
Klaus Weihrauch, Computable Analysis, Springer, Berlin, 2000.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Ishihara, H. (2009). Relativization of Real Numbers to a Universe. In: Lindström, S., Palmgren, E., Segerberg, K., Stoltenberg-Hansen, V. (eds) Logicism, Intuitionism, and Formalism. Synthese Library, vol 341. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8926-8_9
Download citation
DOI: https://doi.org/10.1007/978-1-4020-8926-8_9
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-8925-1
Online ISBN: 978-1-4020-8926-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)