20 Facts About Set theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.

Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice.

Besides its foundational role, set theory provides the framework to develop a mathematical theory of infinity, and has various applications in computer science, philosophy and formal semantics.

Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

Set theory was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers".

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Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" resulting from the power set operation.

Momentum of set theory was such that debate on the paradoxes did not lead to its abandonment.

Set theory is commonly used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation.

Set theory begins with a fundamental binary relation between an object and a set.

Set theory is pure if all of its members are sets, all members of its members are sets, and so on.

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams.

Axiomatic set theory was originally devised to rid set theory of such paradoxes.

Set theory is a promising foundational system for much of mathematics.

However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present.

Set theory is a major area of research in mathematics, with many interrelated subfields.

Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces.

In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending it to make it more broadly applicable.

An inner model of Zermelo–Fraenkel set theory is a transitive class that includes all the ordinals and satisfies all the axioms of ZF.

The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel Set theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]".

Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory.