Relativization of Real Numbers to a Universe

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


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.


Real Number Natural Number Elementary Function Cauchy Sequence Closure Condition 
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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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