Proof sketch for Gödel's first incompleteness theorem

Imagine you're planning a road trip with friends. You want to know if your car will make it to the next gas station before running out of fuel. You can't predict every twist and turn, but you can check the distance and fuel efficiency.

Gödel's theorem and Turing's Halting Problem both show that there are limits to what can be predicted or solved. For Gödel, there are true statements that can't be proven within a theory. For Turing, there's no universal way to know if a computer program will stop running.