Transfinite Set Theory Used in Current-Day Mathematics

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TRANSFINITE SET THEORY USED IN CURRENT DAY MATHEMATICS
Introduction
A set theory is a mathematical theory involving a collection of elements of the set. A set is finite if it contains a finite number of elements or if it’s empty. Cantor, the founder of set theory, explained the one-to-one correspondence between the elements of two sets that have the same cardinality. He also developed the Continuum Hypothesis, which argues that an infinite set of real numbers have a minimal possible greater cardinality than a set on natural numbers since there are more real numbers than natural numbers. Therefore a set of real numbers and a set of natural numbers have the transfinite cardinality (Cohen 89). A Zermelo-Fraenkel set theory consists of a system of axioms. According to Cohen, the continuum hypothesis cannot be proven by the Zermelo-Fraenkel axiom of choice (ZFC). He develops a model that involves forcing, which is a common tool in set theory (Cohen 1143). However, the Continuum hypothesis cannot be proven from the ZFC; hence it is independent of ZFC.
Set theory is applied in various branches of mathematics, most especially the Zermelo-Fraenkel axiom of choice, which leads to the development of other theories. It lays a foundation for modern mathematics as it contains many interrelated research fields. Set theory is a common language of mathematics and is important in understanding the infinite elements, which build up the mathematical structure. There are various transfinite set theory applications outlined below.
Boolean algebra
Boolean algebra is a data structure containing a set, a binary operation that is a union and intersection, a unary operation, and two distinct elements. The binary operations contain two axioms; commutative and associative. This set of axioms are independent, and the union, intersection, and complementation are under systematic differences. Thus, any set of sets closed under the ...