Modern Mathematical Logic

Many of our powerful results about first-order logic, such as the Löwenheim–Skolem Theorem and the Łoś-Vaught Test, focused on countable structures in countable languages. Now that we have a well-developed theory of infinite cardinalities, we can extend these results into the uncountable realm. In addition to the satisfaction we obtain through such generalizations, we will be able to argue that some other important theories are complete, and further refine our intuition about the inability of first-order logic to delineate between infinite cardinalities.