first incompleteness theorem

The first incompleteness theorem states that in any consistent mathematical system powerful enough to include basic arithmetic, there will always be true statements that cannot be proven within that system. This means no single set of rules or axioms can capture all mathematical truths; some truths will always lie beyond formal proof. Essentially, there are limits to what mathematics can prove, highlighting that our understanding of numbers and logic is inherently incomplete, even when the system is consistent and well-constructed.