Set theory

Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.

The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.

Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (/ˈrʌsəl/; 18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic and political activist. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had "never been any of these in any profound sense". He was born in Monmouthshire into one of the most prominent aristocratic families in Britain.

In the early 20th century, Russell led the British "revolt against idealism". He is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore, and his protégé Ludwig Wittgenstein. He is widely held to be one of the 20th century's premier logicians. With A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics. His philosophical essay "On Denoting" has been considered a "paradigm of philosophy". His work has had a considerable influence on logic, mathematics, set theory, linguistics, artificial intelligence, cognitive science, computer science (see type theory and type system), and philosophy, especially the philosophy of language, epistemology, and metaphysics.