Computability theory
<mathematics> The area of theoretical computer science concerning what problems can be solved by any computer.
A function is computable if an
algorithm can be implemented which will give the correct output for any valid input.
Since computer programs are
countable but real numbers are not, it follows that there must exist real numbers that cannot be calculated by any program.
Unfortunately, by definition, there isn't an easy way of describing any of them!
In fact, there are many tasks (not just calculating real numbers) that computers cannot perform.
The most well-known is the
halting problem, the
busy beaver problem is less famous but just as fascinating.
["Computability", N.J. Cutland. (A well written undergraduate-level introduction to the subject)].
["The Turing Omnibus", A.K. Dewdeney].
