Kurt Gödel's incompleteness theorem demonstrates that mathematics contains true statements that cannot be proved. His proof achieves this by constructing paradoxical mathematical statements. To see how the proof works, begin by considering the liar's paradox: "This statement is false." This statement is true if and only if it is false, and therefore it is neither true nor false.
Now let's consider "This statement is unprovable." If it is provable, then we are proving a falsehood, which is extremely unpleasant and is generally assumed to be impossible. The only alternative left is that this statement is unprovable. Therefore, it is in fact both true and unprovable. Our system of reasoning is incomplete, because some truths are unprovable.
Gödel's proof assigns to each possible mathematical statement a so-called Gödel number. These numbers provide a way to talk about properties of the statements by talking about the numerical properties of very large integers. Gödel uses his numbers to construct self-referential statements analogous to the plain English paradox "This statement is unprovable."
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Strictly speaking, his proof does not show that mathematics is incomplete. More precisely, it shows that individual formal axiomatic mathematical theories fail to prove the true numerical statement "This statement is unprovable." These theories therefore cannot be "theories of everything" for mathematics.
The key question left unanswered by Gödel: Is this an isolated phenomenon, or are there many important mathematical truths that are unprovable?