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Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic.
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.
In the language of set theory, atomic formulas are of the form x = y or x ∈ y, standing for equality and set membership predicates, respectively. The first level of the Lévy hierarchy is defined as containing only formulas with no unbounded quantifiers and is denoted by Δ 0 = Σ 0 = Π 0 {\displaystyle \Delta _{0}=\Sigma _{0}=\Pi _{0}} . [ 1 ]
The definition of a finite set is given independently of natural numbers: [3] Definition: A set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion order. Definition: a cardinal n is a natural number if and only if there exists a finite set of which the cardinal is n. 0 = Card (∅)
The great variety and (relative) complexity of formulas involving set subtraction (compared to those without it) is in part due to the fact that unlike ,, and , set subtraction is neither associative nor commutative and it also is not left distributive over ,, , or even over itself.
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However, the discovery of paradoxes in naive set theory, such as Russell's paradox, led to the desire for a more rigorous form of set theory that was free of these paradoxes. In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set ...
There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics).
Further, since set theory was seen as the basis for an axiomatic development of all other branches of mathematics, Russell's paradox threatened the foundations of mathematics as a whole. This motivated a great deal of research around the turn of the 20th century to develop a consistent (contradiction-free) set theory.