Whatever "finite set of symbols" humans use to communicate is not the finite set of symbols that form letters or words. Communication isn't discrete in practical sense, it's continuous - any symbol can take not just different meanings, but different shades and superposition of meanings, based on the differences in way it's articulated (tone, style of writing - including colors), context in which it shows, and context of the whole situation.

The only way you can represent this symbolically is in the trivial sense like you can represent everything, because you can use few symbols to build up natural numbers, and then you can use those numbers to approximate everything else. But I doubt it's what Chomsky had in mind.