Not quite. I can represent pi in a finite way, like so: π. That's about as finite as it gets! However, if you fix a finite alphabet beforehand, you will neve

Not quite. I can represent pi in a finite way, like so: π. That's about as finite as it gets! However, if you fix a finite alphabet beforehand, you will never be able to describe all real numbers using strings constructed from this alphabet. You can always add a new symbol to your alphabet to represent some new irrational number, but no matter how many symbols you add you'll never be able to reach "every" irrational number. Put another way, for a given alphabet there's no way to represent every irrational number in a finite way. However, you could have an irrational number that can be finitely represented in one alphabet, but unable to be finitely represented in another.