The Gödel Incompleteness Theorems

Gödel's Incompleteness Theorems reveal that in any consistent formal system powerful enough to describe basic arithmetic, there will always be true statements that can't be proven within that system. Essentially, no matter how comprehensive a mathematical system is, there will always be logical truths it cannot verify. This shows inherent limits in our ability to fully formalize all mathematical truths, implying that some truths exist beyond formal proof, highlighting fundamental boundaries in the pursuit of complete mathematical understanding.