Relativization of real numbers to a universe

Hajime Ishihara

pp. 189-207

in: Erik Palmgren, Krister Segerberg (eds), Logicism, intuitionism, and formalism, Berlin, Springer, 2009

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.