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Background: Russell's paradox

Between 1910 and 1913, Bertrand Russell and Alfred North Whitehead (1861–1947), a British mathematician and philosopher best known for his work in mathematical logic and the philosophy of science, published their three-volume Principia Mathematica (often abbreviated PM). Russell and Whitehead hoped to achieve what Gottlob Frege could not. Gottlob Frege (1848 – 1925) was a German philosopher, logician, and mathematician, understood by many to be the father of analytic philosophy. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics.

Russell and Whitehead sought to banish the paradoxes of naive set theory by employing a theory of types they devised for this purpose. While they succeeded in grounding arithmetic in a fashion, it is not at all evident that they did so by purely logical means. While Principia Mathematica avoided the known paradoxes and allows the derivation of a great deal of mathematics, its system gave rise to new problems.