British mathematician-philosophers Bertrand Russell and Alfred North Whitehead collaborated on their monumental three-volume Principia Mathematica (1913). Setting out to establish mathematics as a branch of logic, they constructed their arguments from the most basic definitions, axioms, and postulates of math and logic. Though their effort was ultimately found to be unsuccessful, the mathematical, philosophical, and scientific communities were abuzz with discussions of their work throughout the decade. Russell's famous paradox — "is there a mathematical set of all sets not self-members?" — was a troubling problem for mathematicians until in the 1920s the Austrianborn mathematician Kurt Godel, inspired in part by Russell's Paradox, proved the most influential theorem of twentieth-century mathematics: Godel's incompleteness theorem.

Sources:

John Dawson Jr., Logical Dilemmas (New York: A&.K Peters, 1995);

Jean Van Heijenoort, ed., From Frege to Godel (Cambridge, Mass.: Harvard University Press,.1958).