The proof is out there

摘要

Many mathematics undergraduates, used to a diet of theorems and proofs, will hold a comforting belief that all mathematical statements can be proved to be true or false, even if some of these proofs have not yet been discovered. There is, however, a remarkable and unsettling truth lurking in the foundations of mathematics: Goedel's first incompleteness theorem. Consider the statement "This sentence is false". Is this statement true or false? If it's true, then it follows from the statement that it's false, whereas if it's false, then it similarly follows that it's true. The statement is in fact a paradox, neither true nor false.