Can Mathematics Be Proved Consistent?

Kurt Gödel's groundbreaking work in 1931 revealed profound limitations in formal mathematical systems, particularly through his first incompleteness theorem. He demonstrated that in any sufficiently complex system containing elementary arithmetic, there exist true statements that cannot be proven within that system. This challenged the notion that all mathematical truths could be derived from a finite set of rules. Gödel's insights not only transformed mathematics but also raised critical questions about the consistency and completeness of mathematical proofs, leading to further exploration in the field.