incompleteness theorems

Incompleteness theorems, discovered by mathematician Kurt Gödel, show that in any consistent system of mathematics powerful enough to include basic arithmetic, there will always be true statements that cannot be proven within that system. This means that no matter how complete and reliable the system seems, there are limits to what it can prove. Essentially, some truths are inherently unprovable using the system's rules, highlighting fundamental boundaries in formal mathematical reasoning.