You likely know what a paradox is. It's when different logically valid trains of thought lead to contradictory conclusions. Logically, paradoxes shouldn't exist, but they do. As such, it behooves me to question: how do we know that a theorem that is proved using logic is actually undoubtedly true, and not just a paradox?
If this is a bit hard to understand, imagine a particularly unobservant society where everyone only checks if something is true. If this society were to take on the 'this statement is false' paradox, they would check if it's true, realise the paradox (if the statement is true, it's false, but it can't be both at the same time, and so it cannot be true), and then conclude that the statement must be false, since it cannot be true and every claim must be either true or false; missing the fact that this is a paradox. I know this paradox is a particularly bad example, but there are many paradoxes within mathematics (mainly to do with infinity and infinitesimals). How do we know that some of the things that are proved aren't actually paradoxes?