Inexpressibility in Exp-Minus-Log
This provides a foundational result for understanding the limits of expression in a system equivalent to Chow's EL numbers, relevant to computability theory.
The paper proves that every number expressible in the Exp-Minus-Log (EML) system is computable, and demonstrates that Chaitin's constant Ω_U, a non-computable real, cannot be expressed in EML, establishing a formal inexpressibility result.
Odrzywołek defined a system Exp-Minus-Log (EML) that reduces all elementary functions over complex numbers down to a constant `$1$', and a single two place function $E(α, β) = \exp(α) - \log(β)$. This paper shows that in this system, equivalent to Chow's EL numbers, every EML-expressible number is computable. We go on to prove that the canonical example of a non-computable real, Chaitin's $Ω_U$, is inexpressible in EML. This gives a formal inexpressibility theorem for this system.