Gödel’s Incompleteness: The #1 Mathematical Breakthrough of the 20th Century

Gödel’s Incompleteness Theorem says:

“Anything you can draw a circle around cannot explain itself without referring to something outside the circle - something you have to assume but cannot prove.”

You can draw a circle around all of the concepts in your high school geometry book. But they’re all built on Euclid’s 5 postulates which we know are true but cannot be proven. Those 5 postulates are outside the book, outside the circle.

Stated in Formal Language:

Gödel’s theorem says: “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”

The Church-Turing thesis says that a physical system can express elementary arithmetic just as a human can, and that the arithmetic of a Turing Machine (computer) is not provable within the system and is likewise subject to incompleteness.

Any physical system subjected to measurement is capable of expressing elementary arithmetic. (In other words, children can do math by counting their fingers, water flowing into a bucket does integration, and physical systems always give the right answer.)

Therefore the universe is capable of expressing elementary arithmetic and like both mathematics itself and a Turing machine, is incomplete.

You can draw a circle around a bicycle. But the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself.

You can draw the circle around a bicycle factory. But that factory likewise relies on other things outside the factory.

Gödel proved that there are ALWAYS more things that are true than you can prove. Any system of logic or numbers that mathematicians ever came up with will always rest on at least a few unprovable assumptions.

Gödel’s Incompleteness Theorem applies not just to math, but to everything that is subject to the laws of logic. Everything that you can count or calculate. Incompleteness is true in math; it’s equally true in science or language and philosophy.

By far, the best "explanation in layman's terms" I've come across yet!