eli5: What is Godel’s Incompleteness Theorem in math and computer science?

Question

What are the problems and theorems that are derived from it?

In: 80

Answer 1

Gödel’s first incompleteness theorem:
the sentence “this sentence is false” is neither true nor false, and you can’t avoid making statements like that in any reasonably complicated system.

Gödel’ second incompleteness theorem:
We can only hope that math is logically consistent with itself and doesn’t lead to self-contradictions. If we ever succeed at proving that mathematics is consistent, then there’s a way to turn that into a self-contradiction, meaning we actually failed. Again, this is a fact about all reasonably complicated ways of setting up the rules of math.