Theory of Incompleteness

The Theory of Incompleteness, developed by mathematician Kurt Gödel, shows that in any logical system powerful enough to include basic arithmetic, there are true statements that cannot be proven within that system. In other words, no system of formal rules can be both complete (able to prove all truths) and consistent (free of contradictions). This reveals inherent limits in our ability to fully capture all mathematical truths through formal methods, highlighting that some truths exist beyond formal proof, underscoring the fundamental boundaries of mathematical logic.