In his doctoral thesis, “Über die Vollständigkeit des Logikkalküls” (“On the Completeness of the Calculus of Logic”), published in a slightly shortened form in 1930, Gödel proved one of the most important logical results of the century—indeed, of all time—namely, the completeness theorem, which established that classical first-order logic, or predicate calculus, is complete in the sense that all of the first-order logical truths can be proved in standard first-order proof systems.
This, however, was nothing compared with what Gödel published in 1931—namely, the incompleteness theorem: “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (“On Formally Undecidable Propositions of Principia Mathematica and Related Systems”). Roughly speaking, this theorem established the result that it is impossible to use the axiomatic method to construct a mathematical theory, in any branch of mathematics, that entails all of the truths in that branch of mathematics. (In England, Alfred North Whitehead and Bertrand Russell had spent years on such a program, which they published as Principia Mathematica in three volumes in 1910, 1912, and 1913.) For instance, it is impossible to come up with an axiomatic mathematical theory that captures even all of the truths about the natural numbers (0, 1, 2, 3,…). This was an extremely important negative result, as before 1931 many mathematicians were trying to do precisely that—construct axiom systems that could be used to prove all mathematical truths. Indeed, several well-known logicians and mathematicians (e.g., Whitehead, Russell, Gottlob Frege,David Hilbert) spent significant portions of their careers on this project. Unfortunately for them, Gödel’s theorem destroyed this entire axiomatic research program.