Finitely Bounded Effective Computability

One of the most important property of the computability is the certainty. For example a set of natural numbers is computable if there is a Turing machine which decides certainly if a given natural number belongs to the set or not, and no wrong answers are tolerant. On the other hand, the finitely bounded computability discussed in this paper allows finitely many mistakes which can be eventually corrected during an effective procedure. We will show especially that the class of finitely bounded computable real numbers has very interesting properties, and it is a real closed field containing even properly in the class of d-c.e. real numbers.