The incompleteness theorems, developed by mathematician Kurt Gödel in the 1930s, demonstrate fundamental limitations in formal mathematical systems. The first theorem states that in any consistent system that is capable of expressing basic arithmetic, there are true statements that cannot be proven within that system. The second theorem goes further, showing that such a system cannot prove its own consistency. This means that no matter how robust a mathematical framework may seem, there will always be truths that lie beyond its reach, highlighting the inherent limitations of formal logic and mathematics.