Empowered by artificial intelligence technologies, computers today can engage in convincing conversations with people (thanks, ChatGPT), compose songs, paint paintings, play chess and go and diagnose diseases, to name just a few examples of their technological prowess.
These successes could be taken to indicate that computation has no limits. To see if that's the case, it's important to understand what makes a computer powerful.
There are two aspects to a computer's power: the number of operations its hardware can execute per second and the efficiency of the algorithms it runs. The hardware speed is limited by the laws of physics. Algorithms — basically sets of instructions — are written by humans and translated into a sequence of operations that computer hardware can execute. Even if a computer's speed could reach the physical limit, computational hurdles remain due to the limits of algorithms.
These hurdles include problems that are impossible for computers to solve and problems that are theoretically solvable but in practice are beyond the capabilities of even the most powerful versions of today's computers. Mathematicians and computer scientists attempt to determine whether a problem is solvable by trying them out on an imaginary machine.