Definitional recursion has to bottom out somewhere. (OK, it can also be circular, but I'm guessing you would not find that satisfactory.) Whatever words I use to define "function" you can always turn around and insist that I define those words. It's a never-ending game. It ultimately boils down to the definitions of words like "true" and "false, "same" and "different", whose meanings can only be communicated by way of examples: X and X are the same, X and Y are different.

But none of this has anything to do with the matter at hand. There are a finite number of atoms in the universe. Those atoms can only arrange themselves into a finite number of sentient creatures (or computers), each of which has only a finite brain in which can reside only a finite number of thoughts. So no matter how you slice it, the number of realized ideas in this universe is going to be not only countable but actually finite because there is only a finite amount of time before heat death.