User blog:Vel!/Symbols function

This is my stab at the idea of a "symbols function," like the one used in Rayo's number and Ikosarakt's blog. Before I launch into defining my own, I'm going to explain what a symbols function exactly is.

Let \(\Lambda\) denote a finite set of symbols. Define a string \(S\) to be a list containing a finite number of \(\Lambda\) symbols, and call a parser \(\Pi(S)\) be a function mapping strings to positive integers. The symbols function of \(\Pi\) is a function \(\Psi : \mathbb{N} \rightarrow \mathbb{N}\) defined as the smallest positive integer \(\Psi(n)\) such that there does not exist a string \(S\) with at most \(n\) symbols such that \(\Pi(S) = n\).