Is there an algorithm to calculate any real number. Given a ∈ R .
Is there an algorithm to calculate any real number. Given .
Answer & Explanation
, the set of real numbers, is uncountable, but every recursively enumerable set is necessarily countable. This immediately implies that there exist uncomputable real numbers. The same argument shows that there are (formally) indescribable real numbers. Indeed, almost all real numbers are indescribable, and a fortiori, uncomputable. There is nothing wrong about this, though it may be disturbing.
Do uncomputable/indescribable real numbers ‘exist’? Well, that's a philosophical question. A Platonist might say that they exist, even though we have no means of naming them specifically. A finitist might say they don't exist, precisely because we have no algorithm to compute or even recognise such a number.
By and large we deal with generic real numbers, not specific ones. For example, the fact that every non-zero real number has an inverse does not rely on the computability properties of .