Principles of Mathematical Logic is the 1950[1] American translation of the 1938 second edition[2] of David Hilbert's and Wilhelm Ackermann's classic text Grundzüge der theoretischen Logik,[3] on elementary mathematical logic.
The 1928 first edition thereof is considered the first elementary text clearly grounded in the formalism now known as first-order logic (FOL).
Hilbert and Ackermann also formalized FOL in a way that subsequently achieved canonical status.
FOL is now a core formalism of mathematical logic, and is presupposed by contemporary treatments of Peano arithmetic and nearly all treatments of axiomatic set theory.
The former problem was answered in the negative first by Alonzo Church and independently by Alan Turing in 1936.