 Cantor’s set theory (which we will not address directly here as it was not formalized) arose out of an analysis of complicated subcollections of the real line defined using tools of what we would now call topology (Cantor 1872). A better advertisement for the usefulness of set theory for foundations of mathematics (or at least one easier to understand for the layman) is Dedekind’s definition of real numbers using “cuts” in the rational numbers (Dedekind 1872) and the definition of the natural numbers as sets due to Frege and Russell (Frege 1884).