A computability theory of real numbers

In mathematics, various representations of real numbers have been investigated. Their standard effectivizations lead to equivalent definitions of computable real numbers. For the primitive recursive level, however, these effectivizations are not equivalent any more. Similarly, if the weaker computability is considered, we usually obtain different weak computability notions of reals according to different representations of real number. In this paper we summarize several recent results about weak computability of real numbers and their hierarchies.