Naive set theory

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics. Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping stone towards more formal treatments. A naive theory in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses natural language to describe sets and operations on sets. The words and, or, if ... then, not, for some, for every are treated as in ordinary mathematics. As a matter of convenience, use of naive set theory and its formalism prevails even in higher mathematics – including in more formal settings of set theory itself. The first development of set theory was a naive set theory. It was created at the end of the 19th century by Georg Cantor as part of his study of infinite sets and developed by Gottlob Frege in his Grundgesetze der Arithmetik. Naive set theory may refer to several very distinct notions. It may refer to Informal presentation of an axiomatic set theory, e.g. as in Naive Set Theory by Paul Halmos. Early or later versions of Georg Cantor's theory and other informal systems. Decidedly inconsistent theories (whether axiomatic or not), such as a theory of Gottlob Frege that yielded Russell's paradox, and theories of Giuseppe Peano and Richard Dedekind. The assumption that any property may be used to form a set, without restriction, leads to paradoxes. One common example is Russell's paradox: there is no set consisting of "all sets that do not contain themselves".