Inexpressibility in Exp-Minus-Log

Abstract

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 $E(\hat{I}\pm ,{\hat{I}}^2)=\exp (\hat{I}\pm )-\log ({\hat{I}}^2)$. 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 $\hat{I}{\text{c◯}}_U$, is inexpressible in EML. This gives a formal inexpressibility theorem for this system.

5 pages