The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy

Consider the following sentence: “This statement is false.” Is that true? If so, that would make this statement false. But if it’s false, then the statement is true. By referring to itself directly, this statement creates an unresolvable paradox.

So if it’s not true and it’s not false—what is it? This question might seem like a silly thought experiment. But in the early 20th century, it led Austrian logician Kurt Gödel to a discovery that would change mathematics forever. Gödel’s discovery had to do with the limitations of mathematical proofs. A proof is a logical argument that demonstrates why a statement about numbers is true.

The building blocks of these arguments are called axioms—undeniable statements about the numbers involved. Every system built on mathematics, from the most complex proof to basic arithmetic, is constructed from axioms. And if a statement about numbers is true, mathematicians should be able to confirm it with an axiomatic proof. Since ancient Greece, mathematicians used this system to prove or disprove mathematical claims with total certainty.

But when Gödel entered the field, some newly uncovered logical paradoxes were threatening that certainty. Prominent mathematicians were eager to prove that mathematics had no contradictions. Gödel himself wasn’t so sure. And he was even less confident that mathematics was the right tool to investigate this problem.

While it’s relatively easy to create a self-referential paradox with words, numbers don't typically talk about themselves. A mathematical statement is simply true or false. But Gödel had an idea. First, he translated mathematical statements and equations into code numbers so that a complex mathematical idea could be expressed in a single number. This meant that mathematical statements written with those numbers were also expressing something about the encoded statements of mathematics.

In this way, the coding allowed mathematics to talk about itself. Through this method, he was able to write: “This statement cannot be proved” as an equation, creating the first self-referential mathematical statement. However, unlike the ambiguous sentence that inspired him, mathematical statements must be true or false. So which is it? If it’s false, that means the statement does have a proof.

But if a mathematical statement has a proof, then it must be true. This contradiction means that Gödel’s statement can’t be false, and therefore it must be true that “this statement cannot be proved.” Yet this result is even more surprising, because it means we now have a true equation of mathematics that asserts it cannot be proved.

This revelation is at the heart of Gödel’s Incompleteness Theorem, which introduces an entirely new class of mathematical statement. In Gödel’s paradigm, statements still are either true or false, but true statements can either be provable or unprovable within a given set of axioms. Furthermore, Gödel argues these unprovable true statements exist in every axiomatic system.

This makes it impossible to create a perfectly complete system using mathematics, because there will always be true statements we cannot prove. Even if you account for these unprovable statements by adding them as new axioms to an enlarged mathematical system, that very process introduces new unprovably true statements. No matter how many axioms you add, there will always be unprovably true statements in your system.

It’s Gödels all the way down! This revelation rocked the foundations of the field, crushing those who dreamed that every mathematical claim would one day be proven or disproven. While most mathematicians accepted this new reality, some fervently debated it. Others still tried to ignore the newly uncovered a hole in the heart of their field.

But as more classical problems were proven to be unprovably true, some began to worry their life's work would be impossible to complete. Still, Gödel’s theorem opened as many doors as it closed. Knowledge of unprovably true statements inspired key innovations in early computers. And today, some mathematicians dedicate their careers to identifying provably unprovable statements.

So while mathematicians may have lost some certainty, thanks to Gödel, they can embrace the unknown at the heart of any quest for truth.