Wikiproofs:Set-theoretical definition of JHilbert concepts/Chapter 1

Names
The individual components of a JHilbert module are identified by names. A name is a finite, non-empty sequence of symbols. We shall denote our set of available symbols by $$\Sigma$$ and our set of names by $$\mathcal{N}$$, so that $$\mathcal{N}=\Sigma^+$$.

For the purpose of this formal description, it is not necessary to specify the set of symbols $$\Sigma$$ beyond the requirement that it must be non-empty. Further specification of $$\Sigma$$ is the task of the JHilbert specification. The non-emptiness of $$\Sigma$$ immediately implies

Proposition 1.1. The set of names $$\mathcal{N}$$ is infinite.

In fact, for the most part of this formal description, we need only this fact. The explicit representation of names as finite sequences of symbols is only required for prefixing during import and export.