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Relativization of Real Numbers to a Universe

  • Hajime Ishihara
Part of the Synthese Library book series (SYLI, volume 341)

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

Real Number Natural Number Elementary Function Cauchy Sequence Closure Condition 
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.

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Hajime Ishihara
    • 1
  1. 1.School of Information ScienceJapan Advanced Institute of Science and TechnologyNomiJapan

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