Gödel’s Incompleteness Theorem

Gödel’s Incompleteness Theorem is very important, and also not easy to understand. Looking through the article on Wikipedia, you will get a feeling of its full depth.

I didn’t know its meaning clearly until very recently. Some easily accessible explanations are very helpful.

“In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms … of that mathematical branch itself. “

It can be inferred

“You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you’ll only create a larger system with its own unprovable statements. “

It has many important implications:

1. Logic

All logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.

2. Computer

A computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths.

3. Human Being

You’ll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.