Set theory Simplified


Georg Cantor born in the 1870s, discovered the set theory when he stumbled upon the fact that the number of points/dots that make up a line cannot be counted using any natural numbers. He came up with the idea that the numbers contained within the set of real numbers is greater than the one contained within the set of natural numbers. Though real numbers and natural numbers are said to be infinite, the cardinality/size of the objects within the set containing real numbers is greater.

The set theory can be called a collection of data in a wrapper called a 'set', the data contained within the set is called the 'members' or 'elements', of that set. Pure set theory is concerned with sets whose member data are also considered sets.

For example: The empty or null set, is considered a pure/hereditary set.

Therefore Pure Set theory is the study of infinite sets. The axioms that make up set theory lean towards the existence of a set-theoretic universe abundant in mathematical objects and can be construed as sets. Thus the set theory has become the foundation for mathematics, every mathematical instance can be considered a set or a member of a set.

A ∈ B is written to express that A is a member of the set B. This is one of the axioms of the set theor.

  • Extensionality: If two sets A and B have the members, they are equal.
  • Null Set: There exists a set, denoted by ∅ and called the empty set, which has no data/members/elements.
  • Pair: Given any sets A and B, there exists a set, denoted by {A,B}, which contains A and B as its only elements.
  • Union: For every set A, there exists a set, denoted by ⋃A and called the union of A, whose elements are all the elements of the elements of A.
  • Power Set: For every set A there exists a set, denoted by P(A) and called the power set of A, whose elements are all the subsets of A.
  • Infinity*:* There exists an infinite set. In particular, there exists a set Z that contains ∅ and such that if A ∈ Z, then ⋃{A,{A}} ∈ Z.
  • Foundation: Every non-empty set A contains an ∈-minimal element, that is, an element such that no element of A belongs to it.
  • Separation: For every set A and every given property, there is a set containing exactly the elements of A that have that property. A property is given by a formula φ of the first-order language of set theory.

It is important to check that these operations satisfy the following properties:

  • Associativity:

    • A ∪ (B∪C) = (A∪B) ∪ C
    • A∩(B∩C) = (A∩B)∩C
  • Idempotency:

    • A ∪ A = A
    • A ∩ A = A
    • A ∪ ∅ = A
    • A ∩ ∅ = ∅
    • A − A = ∅
  • Commutativity:

    • A ∪ B = B ∪ A
    • A ∩ B = B ∩ A
  • Distributivity:

    • A ∪ (B∩C) = (A∪B) ∩ (A∪C)
    • A ∩ (B∪C) = (A∩B) ∪ (A∩C)
  • If A ⊆ B, then

    • A ∪ B = A ∪ (B−A) =B
    • A ∩ B = A

From Georg Cantor till around 1940, set theory developed mostly around the study of the continum, that is, the real line containing said infinite dots. The main topic being the study of the regularity properties, of simply-definable sets of real numbers, an area of mathematics that is known as Descriptive Set Theory.

Descriptive Set Theory is the study of the properties and members of definable sets of real numbers and, more generally, the subsets of the main set.

This is a simplified, quick read version of the set theory. If you guys want a more in depth version of this post, or want a simplified version of any topic, do let me know in the comment section below and I will try my best to fulfill my promist