What is Set Theory?

Set theory is a basic branch of mathematics, and the research object is general sets. Set theory holds a unique position in mathematics, and its basic concepts have penetrated into all areas of mathematics. Set theory or set theory is a mathematical theory that studies sets (a group of abstract objects), and contains the most basic mathematical concepts such as sets, elements, and member relationships. In most modern mathematical formulations, set theory provides a language for describing mathematical objects. Set theory, logic and first-order logic together form the axiomatic basis of mathematics. Mathematical objects are formally constructed in terms of undefined "sets" and "set members".

Set theory is a branch of mathematics that studies the structure, operations, and properties of sets. The most important basic theory of modern mathematics was founded by Cantor in the 1870s and 1980s. A set of points on a plane (or space) is called a "point set". A point set can be some isolated points, or all points on a curve or in an area. Various geometric figures can be regarded as a set of points, and then the common features of the points contained in the position and quantity relations can be studied, which can often lead to deeper conclusions than intuitive. The basic theory of point sets is called point set theory, and set theory discusses general sets that are broader and more abstract than point sets.

Set theory has a wide range of applications in various branches of mathematics such as geometry, algebra, analysis, probability theory, mathematical logic, and programming languages. The elements of a collection should satisfy certain axioms. Various axiomatic systems of set theory can be established. For example, from 1904 to 1908, Zemelo (German, 1871-1953) proposed the first set theory of axiom system (ZF system) to avoid Russell's paradox. Important questions about the basis of set theory have not yet been satisfactorily resolved.

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At first, some mathematicians rejected set theory as the basis of mathematics, thinking that it was just a game with fantasy elements. Erit Bishop dismissed set theory as "God's mathematics and it should be left to God." and,

The equipotentiality principle of set theory was prepared by Cantor in order to build the theoretical and logical basis for modern analysis, not to describe the "common sense world". It is ridiculous to try to use "common sense" to refute equipotential principles. Just as it is meaningless to think about real infinity in real life, because you can only give examples of potential infinity (for example, when exploring the truth, the repetition between practice and knowledge until infinity), but you ca nt mention the infinite example. As long as it can logically form a consistent system, it is the correct foundation under the modern analysis system.

As a constructive principle, Cantor's theoretical assumptions can be replaced, which has been clarified in the study of controversial axioms. However, if some axioms are replaced to form a new system, it can only describe the new system. It cannot describe the original system as "wrong." ^[4]