The 19th century goal was to use set theory as the foundation for mathematics and thereby extend logic to all of science.
Well, in the 19th C., something like that was really only the goal of Dedekind. Frege, of course, had a similar goal, but he attempted to use *logic* as the foundation of mathematics. Granted, his logic included a class theory (sadly, an inconsistent one), but it was fundamental to Frege's project that the axioms of the theory be considered part of logic. The foundational project based in set theory really didn't begin in earnest until the early 20th century after the discovery of Russell's paradox and the consequent axiomatization of set theory by Zermelo.
But in practice, working mathematicians *ignore* the foundational work. It may be a required course, but it doesn't help them solve problems.
Indeed! I think I posted this before — a mathematician friend of mine's reply when I sent him a link about the 100th anniversary of Zermelo's first axiomatization of set theory — but it's worth resending.