Once you've written down the axioms, to decide whether some given statement is true or false, you try to prove it from the axioms. This way of doing mathematics was introduced by the ancient Greeks 2,500 years ago, and worked well throughout history. It was always assumed that the only thing that could prevent you from being able to decide the truth or falsity of any mathematical question that came up was if you'd overlooked one or more basic assumptions to put into your list of axioms. His Incompleteness Theorem says that you can never find enough axioms. No matter how carefully you try to make sure you have written down all the basic assumptions, there will always be some questions that you can't answer.
Mathematical knowledge is destined to remain forever incomplete.
In fact, the situation is even worse. You can never be sure that, in writing down your axioms, you haven't made a mistake and introduced some subtle conflict. To some extent, that's still true. Pythagoras's theorem about right-triangles is as true today as it was when it was first proved 3,000 years ago, and it will remain true forever. Once a mathematician has proved something, it's truth is permanent.