COMPUTING STRENGTH OF STRUCTURES RELATED TO THE FIELD OF REAL NUMBERS

Abstract

AbstractIn [8], the third author defined a reducibility $\le _w^{\rm{*}}$ that lets us compare the computing power of structures of any cardinality. In [6], the first two authors showed that the ordered field of reals ${\cal R}$ lies strictly above certain related structures. In the present paper, we show that $\left( {{\cal R},exp} \right) \equiv _w^{\rm{*}}{\cal R}$. More generally, for the weak-looking structure ${\cal R}$ℚ consisting of the real numbers with just the ordering and constants naming the rationals, all o-minimal expansions of ${\cal R}$ℚ are equivalent to ${\cal R}$. Using this, we show that for any analytic function f, $\left( {{\cal R},f} \right) \equiv _w^{\rm{*}}{\cal R}$. (This is so even if $\left( {{\cal R},f} \right)$ is not o-minimal.)