In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history.Lubos Motl liked the essay.
Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths.
But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency.
His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.
This description of Goedel's theorems is common, but I doubt that Goedel would agree with it.
In spite of the above, ZFC set theory is a perfectly good foundation for mathematics. It is logical, consistent, and good enough to prove all the theorems in your favorite math textbooks.
Yes, ZFC is incomplete in the sense that you cannot use it to program a computer to answer any mathematical question. Life is not so simple.
This reminds me a little bit of Einstein's complaints that quantum mechanics is incomplete. Somehow the theory is good enough to explain the physics underlying about a trillions dollars worth of global production, but somehow it does not answer every question in the way that Einstein thought should be answered.
Supposedly Hilbert was the one who wanted an axiom system that was consistent and complete. I am not sure he did. He did say he wanted an axiom system that was demonstrably consistent, but I don't think he ever said that the consistency proof should be within the axiom system.
If he did, then he made a minor misstatement of what was possible. But the larger goal of axiomatizing mathematics has been very successful, and Hilbert and Goedel played roles in that.
All these articles saying Mathematics and Physics have faulty foundations are wrong.