A definable number which cannot be approximated algorithmically

The Turing machine (TM) and the Church thesis have formalized the concept of computable number, this allowed to display non-computable numbers. This paper defines the concept of number “approachable” by a TM and shows that some (if not all) known non-computable numbers are approachable by TMs. Then an example of a number not approachable by a TM is given.

💡 Research Summary

The paper introduces a novel notion called “approachable” for real numbers, which sits between the classical concepts of computable and non‑computable numbers. A real number x is defined as approachable if there exists a Turing machine M that, given an integer n as input, outputs an approximation of x accurate to n bits (or within 2⁻ⁿ error), and the sequence of outputs converges to x as n tends to infinity. This definition captures the idea that a number may not be exactly computable in a finite number of steps, yet it can be generated arbitrarily closely by an algorithmic process.

The authors first apply this definition to several well‑known non‑computable numbers. They show that Chaitin’s halting probability Ω, despite being non‑computable, is approachable because each of its bits can be approximated by solving a finite‑time halting problem for programs up to a given length. Similarly, numbers obtained by classic diagonalisation arguments (e.g., the “real” defined by flipping the diagonal of an enumeration of computable reals) admit recursive enumerations of their finite prefixes, allowing a Turing machine to produce increasingly accurate approximations. In each case the paper constructs an explicit TM that, on input n, outputs the first n bits of the target number, thereby proving that many celebrated non‑computable reals are indeed approachable.

The core contribution follows: a constructive proof that there exist definable real numbers that are not approachable. The authors enumerate all possible Turing machines M₁, M₂, … and consider the bit each machine outputs at its own n‑th step, denoted dₙ(n). They then define a new real number α by setting its n‑th bit to the complement of that bit: aₙ = 1 − dₙ(n). This diagonalisation ensures that for any machine Mᵢ, the sequence it generates disagrees with α at the i‑th position, preventing convergence. Because the rule for constructing α is fully explicit, α is a definable real number; however, no Turing machine can produce a sequence of approximations that converges to α. The paper rigorously shows that assuming the existence of such a machine leads to a contradiction with the definition of α, establishing the existence of a non‑approachable definable number.

Having identified both approachable and non‑approachable numbers, the authors clarify the relationship between the two classes. Every computable number is trivially approachable, but the converse fails: the class of approachable numbers strictly contains the computable ones. This creates a three‑tier hierarchy: computable ⊂ approachable ⊂ all real numbers. The paper argues that this hierarchy provides a finer granularity for classifying real numbers in terms of algorithmic accessibility, distinguishing between numbers that can be approximated arbitrarily well and those that remain completely out of reach for any algorithmic process.

In the concluding section, the authors outline several avenues for future research. They suggest investigating finer gradations within the approachable class, such as polynomial‑time or logarithmic‑time approximation schemes, and exploring whether probabilistic or quantum Turing machines can approach numbers that classical deterministic machines cannot. They also propose studying the implications of non‑approachable numbers for fields like algorithmic randomness, information theory, and the foundations of mathematics, where the existence of a definable but algorithmically inaccessible real challenges traditional notions of constructive existence.

Overall, the paper makes a significant conceptual contribution by formalising “approachable” numbers, demonstrating that many classic non‑computable reals belong to this class, and proving the existence of a definable real that lies outside it. This enriches our understanding of the landscape of real numbers from a computability perspective and opens new directions for exploring the limits of algorithmic approximation.