Can anyone find a mapping from the set of all possible string to the natural numbers?
Just pretend that your alphabet $S = \{0, \dots, s\}$ for some&nbsp;$s$ and then a string in&nbsp;$S^*$ is just a natural number written out in base $s+1$. There's the slight wrinkle that $0$, $00$, $000$, etc. are all representations of zero, but that's easily fixed by associating the string $x$ with the number whose base-$(s+1)$ representation is&nbsp;$1x$ (i.e., $x$&nbsp;with a&nbsp;$1$ stuck on the front).