From the Introduction. In 1931 there appeared in a German scientific periodical a relatively short paper with the forbidding title Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme ( On Formally Undecidable Propositions of Principia Mathematica and Related Systems ). Its author was Kurt Godel, then a young mathematician of 25 at the University of Vienna and since 1938 a permanent member of the Institute for Advanced Study at Princeton. The paper is a milestone in the history of logic and mathematics. When Harvard University awarded Godel an horary degree in 1952, the citation described the work as one of the most important advances in logic in modern times. At the time of its appearance, however, neither the title of Godel's paper r its content was intelligible to most mathematicians. The Principia Mathematica mentioned in the title is the monumental three-volume treatise by Alfred North Whitehead and Bertrand Russell on mathematical logic and the foundations of mathematics; and familiarity with that work is t a prerequisite to successful research in most branches of mathematics. Moreover, Godel's paper deals with a set of questions that has never attracted more than a comparatively small group of students. The reasoning of the proof was so vel at the time of its publication that only those intimately conversant with the technical literature of a highly specialized field could follow the argument with ready comprehension. Nevertheless, the conclusions Godel established are w widely recognized as being revolutionary in their broad philosophical import. It is the aim of the present essay to make the substance of Godel's findings and the general character of his proof accessible to the n-specialist. Godel's famous paper attacked a central problem in the foundations of mathematics. It will be helpful to give a brief preliminary account of the context in which the problem occurs. Everyone who has been exposed to elementary geometry will doubtless recall that it is taught as a deductive discipline. It is t presented as an experimental science whose theorems are to be accepted because they are in agreement with observation. This tion, that a proposition may be established as the conclusion of an explicit logical proof, goes back to the ancient Greeks, who discovered what is kwn as the axiomatic method and used it to develop geometry in a systematic fashion. The axiomatic method consists in accepting without proof certain propositions as axioms or postulates (e.g., the axiom that through two points just one straight line can be drawn), and then deriving from the axioms all other propositions of the system as theorems. The axioms constitute the ''foundations of the system; the theorems are the superstructure, and are obtained from the axioms with the exclusive help of principles of logic.