- Author
- Natalie Wolchover, Quanta Magazine
- Published
- 14 July 2020
- Kind
- Magazine explainer on a mathematical proof, with worked examples
- Original
- www.quantamagazine.org/how-godels-proof-works-20200714/
- Subject
- Mathematics and optimization
- Themes
- Checking the work, Getting good
What it is
Natalie Wolchover is a columnist at Quanta Magazine. The article is an explainer. It shows how Gödel mapped statements about axioms onto statements about numbers.
What it says
- The article says any consistent axiom set is incomplete, with true unprovable facts.
- It says Gödel numbering gives each formula a unique integer through prime factorization.
- It says statements about formulas can be turned into arithmetic formulas with their own numbers.
- It says the formula G asserts of itself that it cannot be proved.
- It says G is true yet undecidable, and new axioms just yield a new G.
- It says no consistent axiom set can prove its own consistency, the second theorem.
- It says undecidable questions have since appeared, such as the halting problem.
What evidence it gives
The article gives worked examples. One example encodes the formula 0 = 0 as the number 243,000,000. It then walks through the self-reference step by step. It cites Nagel and Newman's 1958 book. It gives no formal proof.
Interests and limits
Quanta is funded by the Simons Foundation and is editorially independent. The account is simplified and informal, as the article says. It skips technical details, so it cannot replace the original proof. The remark about physics is stated only as a suggestion.