The most famous example is the halting problem. It supposes there was a program that could take another program and input for that program as its own input, and then output if the program given to it will halt (produce an output or otherwise stop running) when given that input, or if it will get stuck in some kind of loop forever. There’s a proof that shows this problem is undecidable and that you cannot have a general program that can tell if any arbitrary programs will halt on any arbitrary input. The Collatz conjecture is a famous unproven statement in math that could be true but undecidable so we’ll never actually know.
This is getting away from lying which requires intent but whatever