The Infinite Regress of Human Vanity
In 1931 the Czech-born Kurt Godel, in what came to be known as the Incompleteness Theorems, demonstrated that within any mathematical system there are true propositions that cannot be proved or disproved on the basis of the rules and axioms within that system. You might be able to prove every conceivable mathematical statement by going outside the system and coming up with new rules and axioms, but by doing so you’ll only create a larger system with its own non-provable propositions. And so on and so on ad infinitum. Thus illustrating that all logical systems of any complexity are by definition incomplete; each contains more true statements than it can possibly prove according to its own defining set of rules and axioms.
Godel’s theorem has been used to argue that you can never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.