If we have a finite alphabet, then the set of programs we can write is countably infinite (aleph naught). The set of all functions is uncountably infinite (cardinality of real numbers).
If we have an infinite alphabet, does that change the cardinality of the set of programs we can write? Can we write programs for all functions then?
Intuitively, if I tried to count the number of such programs, I would get stuck at the first character and never progress to the second character. That seems to suggest that it's uncountably infinite, but then I remember Cantor's trick of presenting rational numbers in diagonal rows, so I'm not sure if there is a (perhaps similar) way of presenting the programs in a countable way as well.