SET THEORY
What is set theory?
Set theory is the mathematical theory of sets and is associated with logic; it is also
considered the study of sets (collections of objects or entities that can be real objects
or intellectual concepts) and their properties. (For more about sets, see below.) Under
formal set theory, three primitives (undefined terms) are used: S(the set), I(the iden-
tity), and E(the element). Thus, the formulas Sx, Ixy, Exymean “x is a set,” “x is iden-
tical to y,” and “x is an element of y,” respectively.Overall, set theory fits in with the aims of logic research: to find a single formula
theory that will unify and become the basis for all of mathematics. As it turns out, sets
lead directly to a vast amount of data encompassing all of modern mathematics. There
are also a number of different set theories, each having its own rules and axioms. No
matter what version, set theory is not only important to mathematics and logic but
also to other fields, such as computer technology and atomic and nuclear physics.What are naiveand axiomatic set theory?
The naive set theory is not one that takes everything for granted. It is actually a
branch of mathematics that attempts to formalize the nature of the set using the
fewest number of independent axioms possible. But it is not the answer to formalizing
sets, as it quickly leads to a number of paradoxes. Because of this, mathematicians use
a more formal theory called the axiomatic set theory,which is a version that uses
axioms taken as uninterpreted rather than as formalization of preexisting truths. (For
more about axiomatic systems, see elsewhere in this chapter).What is Russell’s Paradox?
Russell’s Paradox is one of the most famous of the set theory paradoxes. It first appears
when studying the naive set theory: In this case, Ris the set of all sets that are not
members of themselves; from there, Ris neither a member of itself nor not a member
of itself. The paradox sets becomes evident when one tries to reason how a set appears
to be a member of itself if and only if it is not a member of itself.Discovered by Welsh mathematician and logician Bertrand Arthur William Russell
(1872–1970) in 1901, the paradox sparked a great deal of work (and controversy) in
logic, set theory, and especially in philosophy and foundations of mathematics. The
reason why it became so important was its effect on mathematics: It created problems
for those who based mathematics on logic, and it also indicated that something was
wrong with Georg Cantor’s intuitive set theory. (For more about Russell and the para-
122 dox, see “History of Mathematics.”)