Inexpressibility in Exp-Minus-Log

TL;DR

This paper proves that while all EML-expressible numbers are computable, certain well-known non-computable numbers like Chaitin's Omega are inexpressible within the Exp-Minus-Log system, establishing a formal inexpressibility result.

Contribution

It demonstrates that the Exp-Minus-Log system cannot express Chaitin's Omega, providing a formal inexpressibility theorem for this computational framework.

Findings

All EML-expressible numbers are computable.

Chaitin's Omega is inexpressible in EML.

The paper establishes a formal boundary for EML's expressiveness.

Abstract

Odrzywo\l{}ek 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 $E(\alpha, \beta) = \exp(\alpha) - \log(\beta)$. 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 $\Omega_U$, is inexpressible in EML. This gives a formal inexpressibility theorem for this system.