Are there real numbers that cannot be uniquely expressed with a finite number of symbols? Is this the same thing as an uncomputable number?
I can show that if there is one such number then there are infinitely many, and the set of all these numbers is not compact if it is nonempty, but I don't know if any such numbers exist.
Suggestions for tags are welcome.
Edit: I want to add that by "symbols" I mean numbers, mathematical symbols, or English language descriptions (or any language really). So for π, you could write it as "the ratio of a circle's circumference to its diameter" and that would count.