The Incompleteness Theorems, formulated by mathematician Kurt Gödel in the 1930s, reveal fundamental limitations in formal mathematical systems. The first theorem states that in any consistent system powerful enough to describe basic arithmetic, there are true statements that cannot be proven within that system. This means that no matter how comprehensive a mathematical framework is, it will always have some truths that remain unprovable. The second theorem goes further, showing that such a system cannot prove its own consistency. In simpler terms, if a system can prove its own reliability, it is likely inconsistent. These theorems have profound implications for mathematics, logic, and the philosophy of mathematics.