On subrecursive representability of irrational numbers

Abstract

We consider various ways to represent irrational numbers by subrecursive functions: via Cauchy sequences, Dedekind cuts, trace functions, several variants of sum approximations and continued fractions. Let S be a class of subrecursive functions. The set of irrational numbers that can be obtained with functions from S depends on the representation. We compare the sets obtained by the different representations.

On General Sum Approximations of Irrational Numbers

“…It was proved in [5] that S g↑ ∩ S g↓ contains exactly the irrational numbers that have a continued fraction in the class S. In this paper we prove that S g↑ = S g↓ . Moreover, we prove that…”

“…It is also obvious that there is an elementary function ψ(q, i, b) that yields the i th digit in the base-b expansion of the rational number q. More on elementary functions, primitive recursive functions and honest functions can be found in Section 2 of [5] and in [7].…”

“…It is proved in [5] that we have S g↓ ∩ S g↑ = S T = S [ ] for any S closed under primitive recursive operations. It is conjectured in [5] that S g↓ = S g↑ . Theorem 12 shows that this conjecture holds.…”

“…If we work with some restricted notion of computability, e.g., polynomial time computability or primitive recursive computability, they do not. This phenomenon has been investigated over the last seven decades by Specker [13], Mostowski [8], Lehman [10], Ko [3,4], Labhalla & Lombardi [9], Georgiev [1], Kristiansen [5,6] and quite a few more. Georgiev et al [2] is an extended version of the current paper.…”

“…Irrational numbers can be represented by infinite sums. Sum approximations from below and above were introduced in Kristiansen [5] and studied further in Kristiansen [6]. Every irrational number α between 0 and 1 can be uniquely written as an infinite sum of the form k i ∈ N \ {0} and k i < k i+1 .…”