Gentzen's consistency proof

Gentzen's consistency proof demonstrates that a fundamental system of mathematics, called Peano arithmetic, cannot produce contradictions, like proving both a statement and its opposite. He did this by showing that proofs within this system can be translated into a special kind of well-ordered structure called ordinal numbers, which always exist in a correct system. Since these ordinals are consistent and well-founded, this assures that the original arithmetic system also does not lead to contradictions. In essence, Gentzen provided a way to confirm the reliability of basic mathematical reasoning by linking it to a well-understood and reliable mathematical concept.