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.
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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
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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
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