What do Gödel's incompleteness theorems actually say?

Gödel's two incompleteness theorems concern formal systems, meaning sets of axioms plus rules for deriving conclusions. The first says that any such system that is free of contradictions and can do a certain amount of elementary arithmetic contains statements that can be neither proved nor disproved from within it. The second says that such a system cannot prove its own consistency. Together they ended a hope that mathematics could be fully captured by a finite set of axioms. [1] [2]

In the early twentieth century David Hilbert hoped that all of mathematics could rest on a small, elegant set of self-evident axioms. He wanted the resulting system to be sound, so that only true statements could be proved, and consistent, so that it contained no contradictions. He also wanted it to be complete, so that every statement could be either proved or disproved. Gödel's results showed that a system cannot have all of these properties if it includes a modest amount of arithmetic. The Guardian notes that the result was a huge shock to a community that had been unshakeably optimistic about the reach of mathematics. [1] [3]

The first theorem has three conditions. The system must be consistent, meaning it never proves both a statement and its negation. It must be able to express basic arithmetic, such as addition, multiplication and statements about numbers. Given those, the system is incomplete: some statement in its language is undecidable, neither provable nor disprovable from its axioms. If a system were inconsistent, it could prove anything, which would make it useless, so the theorem is stated only for consistent systems. [4] [5]

Spencer Greenberg gives an informal sketch of why. Gödel showed that a self-referring statement can be built in any such system, saying in effect that it cannot be proved. If the system could prove it, the system would be proving something that says it is unprovable, which would be a contradiction. Since the system is assumed consistent, the statement cannot be proved, which is exactly what it claims, so it is true. Greenberg stresses that being unprovable is not the same as being false. Adding axioms to prove such a statement only produces a bigger system with its own unprovable truths. [6]

The second theorem says that a consistent system with enough arithmetic cannot prove its own consistency. To show the system is free of contradictions, one must step outside it, for example by relying on assumptions that are not among its axioms. Greenberg's explanation links it to the first theorem. A proof of consistency inside the system would also yield a proof of the self-referring statement, which the first theorem rules out. Put another way, a system that proves its own consistency must actually contain a contradiction. [1] [6]

The theorems have profound consequences for the philosophy of mathematics, but they do not cast doubt on the calculations used to build cars and planes or to test medicines. They also do not concern everyday maths such as managing finances. The undecidable statements are typically very complicated ones, and the theorems do not apply to systems too weak for arithmetic. [1] [5] [6]