Gödel's incompleteness theorems

Gödel's incompleteness theorems reveal fundamental limits in formal mathematical systems. The first theorem states that in any consistent system capable of expressing basic arithmetic, there are true statements that cannot be proven within that system. The second theorem shows that such a system cannot prove its own consistency. Essentially, these results demonstrate that no single set of rules can fully capture all mathematical truths, highlighting inherent limitations in formal logical frameworks used to understand mathematics and logic.