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Gödel's incompleteness theorems
In a formal system, there are axioms. All axioms are assumed to be true, at any rate for the purpose of the logic. A theorem then comes up with other true statements from the axioms, using certain rules. A sequence of such statements is called a proof of a statement, because it shows that the statement is true, given the axioms.
Ideally, it should be possible to construct all true statements in the formal system in that manner. A system that has this property is called complete; one that does not is called incomplete. Another thing wanted of a theory is that there should be no contradictions. This means that it is not possible to prove that a statement is true and false at the same time. A system that does not include theories that allow this is called consistent.
Gödel said that every non-trivial formal system is either incomplete or inconsistent: 
- For a given (non-trivial) formal system, there will be statements that are true in that system, but which cannot be proved to be true inside the system.
- If a system can be proved to be complete using its own logic, then there will be a theorem in the system that is contradictory.
Most people think this shows that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible. This would give a negative answer to Hilbert's second problem.