On Signifiable Computability: Part III: A Note on Unnameable Functions on Natural Numbers

A writing system W on an alphabet A is a tuple (R,M), where R is a finite set of text formation rules and M is a finite rule application mechanism that generates texts on A. A natural writing system forms natural language texts on an alphabet such as Sanskrit on Devanagari. An artificial writing system generates formal language texts on an alphabet such as Lisp on Unicode. Let N={0,1,2,…} be the set of natural numbers. A function on natural numbers f:Nk↦N, 0<k∈N, is nameable by a writing system on an alphabet if, and only if, the system can generate a text on the alphabet that names f and no other function. We show that there exist functions on natural numbers unnameable in principle in that they cannot be named by any writing system on any alphabet. Our results imply the following computability-theoretic hierarchy of functions on natural numbers: computable⊊partiallycomputable⊊nameable⊊F, where F is the set of functions on N.