Godel's Incompleteness Theorems

Godel’s Incompleteness Theorems show that in any logical system powerful enough to describe mathematics, there will always be true statements that cannot be proved within that system. Essentially, no system can be both completely consistent and fully capable of proving all mathematical truths. This reveals inherent limits in our ability to formalize all mathematical knowledge, meaning some truths are beyond formal proof, and we must accept a level of uncertainty or incompleteness in mathematical systems.