Structure of Reliable Methods. Characterization theorems tell us how the structure of reliable methods is attuned to the structure of the hypotheses under investigation. For example, the theorem mentioned establishes a connection between falsifiability and testability, but one that is more attenuated than the naïve Popperian envisions: it is not necessary that the hypotheses under test be directly falsifiable; rather, there must be ways of strengthening each hypothesis that yield a countable number of refutable “subhypotheses”. We can think of these refutable subhypotheses as different ways in which the main hypothesis may be true. (For example, one way in which “all but finitely many swans are white” is true is if there are are at most 10 black swans; another is if there are at most 100 black swans, etc.). Strengthening the original hypotheses so they become empirically refutable matches the spirit of  Lakatos’s methodology  in which a general scientific paradigm is articulated with auxiliary hypotheses to define testable (i.e., falsifiable) claims.