Solovay's Theorem

Solovay's Theorem states that if you assume the existence of a certain powerful mathematical universe called an "inaccessible cardinal" (a special kind of infinity), then you can create a consistent system where every statement about real numbers (like questions about infinite decimal expansions) that can be decided as true or false using standard logical rules can also be decided using a specific, well-understood procedure (called the "Solovay model"). In essence, it links deep properties of infinity with the ability to determine the truth of many questions in the realm of real numbers, showing that some complex and independent propositions are actually resolvable under certain assumptions.