Set Theory
Set theory is a mathematical theory of collections, "sets," and collecting, as governed by axioms. Part of its larger significance is that mathematics can be reduced to set theory, with sets doing the work of mathematical objects and their collections and set-theoretic axioms providing the basis for mathematical proofs. With this reduction in play, modern set theory has become an autonomous and sophisticated research field of mathematics, enormously successful at the continuing development of its historical heritage as well as at analyzing strong propositions and gauging their consistency strength.
Set theory arose in mathematics in the late nineteenth century as a theory of infinite collections and soon became intertwined with the development of analytic philosophy and mathematical logic. The subject was then developed as the logical distinction was being clarified between "falling under a concept," to be transmuted in set theory to "x ∈ y", x is a member of y, and subordination or inclusion, to be transmuted in set theory to "x ⊆ y", x is a subset of y. That set theory is both a field of mathematics and serves as a foundation for mathematics emerged early in this development.
In what follows, set theory is presented as both a historical as well as an epistemological phenomenon, driven forward by mathematical problems, arguments, and procedures.
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