Fallibility is the idea that people make mistakes, and there is no foolproof way to avoid making mistakes. There are proofs that fallibility is true, but they're a little boring.
The problem with the proofs is that they show fallibility is strictly true in a technical sense. What's a lot more interesting is the spirit of fallibility: mistakes are common.
There's no hard proof that mistakes are common. But they are. People mess stuff up all the time. It's common knowledge. And confidence, or feeling sure of yourself, is not much of a defense against this. People think they know what they're doing, but still mess up.
Expertise helps more. Experts get good at stuff and make fewer mistakes. But even experts make mistakes pretty often.
The reason science works so well isn't that scientists never mess up. It's because they double check their own work, and other people's work, all the time. Scientists even include sections about possible sources of error in their papers because they recognize their fallibility. Their vigilance helps a lot. And they are motivated to remain vigilant because they understand that otherwise there would be a lot of mistakes going unnoticed.
It's not just scientists who should be vigilant against mistakes. It's everyone.