I'm wondering if there's a way to symbolically (or is there a more lose constraint?) represent ANY irrational number in a systematic way.
You can represent any rational number as two integers and I know you can represent some irrational numbers as sums of some infinite series. Is there any representation that can surely be found for any irrational number?
Now, I'm pretty sure there's an interesting observation that since the set of all possible English sentences is countable, there has to be a great set of numbers that can't be expressed as an English sentence. What I'm looking for is perhaps a way to loosen that rule to allow an uncountable infinity of numbers to be expressed.
My previous research into similar topics has usually led to people saying a question like this is ambiguous because a "representation" is not a very well defined idea. So take your best guess at what the definition might be so that this question makes sense (if there are any).