# Set theory

Set theory is a branch of mathematics dealing with collections of objects, called sets. It revolutionized mathematics and made possible enormous new insights. The approach of set theory also offers a powerful way to think about life, handle anxiety, escape addiction, and understand the miracles in the New Testament. Problems, which otherwise seem impossible to overcome, can be more easily solved or managed with a set-theory approach.

The devoutly Christian founder of set theory, Georg Cantor, viewed it as bringing freedom to mathematics. He summarized set theory as follows: "A set is a Many that allows itself to be thought of as a One." David Hilbert, a leading mathematician in the early 20th century, described set theory as follows:

 “ No one shall drive us from the paradise which Cantor has created for us. ”

John von Neumann, considered by many to have been the most brilliant man of the 20th century (and a convert to Catholicism, did his best work in set theory. He developed a definition of an ordinal number as the set of all smaller ordinal numbers, the definition that is commonly accepted today. That approach helps circumvent issues caused by the transfinite numbers of Georg Cantor.

The language of set theory is based on a single fundamental relation, called membership. We say that is a member of (in symbols ), or that the set contains as an element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements, or, equivalently, if each is a subset of the other.

## History of set theory

It was developed in the late 1800s, primarily by the German mathematician Georg Cantor. This initial attempt became known as "naive set theory" because mathematicians found flaws in it. It was replaced by "axiomatic set theory" in the early 1900s. The most commonly used such axiomatization is Zermelo-Fraenkel set theory.

An initial insight of set theory, against intense opposition by established mathematicians, was that some infinities are larger than others. Previously it was thought that infinity had only one size.

One paradox in naive set theory was announced by Bertrand Russell in 1901, and is known as Russell's Paradox.

Like all sufficiently strong mathematical theories, set theory is incomplete, as shown by Kurt Gödel. However, set theory is the received axiomatization of mathematics today, with subjects like analysis, algebra, topology, and geometry using set theory and its language for their own foundation.

## The empty set

The empty set is the set with no members. Because sets are uniquely defined by membership, the empty set is unique. The empty set is usually denoted by {} or ∅.

In the axiomatic set theory, any other set is build over the empty set. An example of a set with one element is the set { ∅ } - the set whose single element is the empty set. Von Neumann called the empty set 0, this set { ∅ } = 1, and proceeded in defining each natural number n as the set with the n elements from 0 to n-1. In this way, it's possible to define the (infinite) set of natural numbers. Ordered pairs, functions, relations, etc, are all defined as sets with some properties.