Set Theory

Set theory is a branch of mathematics that studies sets, which are essentially collections of objects. For example $\{1,2,3\}$ is a set, and so is $\{\heartsuit, \spadesuit\}$. Set theory is important mainly because it serves as a foundation for the rest of mathematics--it provides the axioms from which the rest of mathematics is built up.

The most straightforward way to define a set is to say that it is an unordered collection of objects. These objects are called the elements of the set, and they can be organized into smaller collections called subsets. Some common examples of sets include $\mathbb{R}$, the set of all real numbers, $\mathbb{Z}$, the set of all integers, and the Cantor set.

Unfortunately, these straightforward definitions can run into problems. For example, because sets are themselves objects, sets can contain other sets--the set $\big\{\{1,2\}, 1, 2\big\}$ contains both the numbers 1 and 2, as well as the set $\{1,2\}$. But this quickly gives rise to Russell's paradox: if $M$ is the set of all sets that do not contain themselves, does $M$ contain itself? By definition, if $M$ did not contain itself, then $M$ would contain itself, leading to a contradiction. Thus, the simple definitions of sets are largely insufficient.

The problems with defining sets led mathematicians to develop various systems for formalizing set theory. The most popular of these is ZFC, where the Z and F stand for Zermelo and Frankel, two mathematicians who developed these axioms, and the C stands for the axiom of choice, one of the axioms involved. This system of set theory provides a rigorous basis for the rest of mathematics but can lead to some unintuitive results.

In particular, the axiom of choice can give rise to a variety of paradoxes, including the Banach-Tarski paradox, in which a single ball can be cut into pieces, and reassembled into two balls, each of which is the same size as the original. On the other hand, many intuitive and important statements also follow from the axiom of choice, such as the fact that every vector space has a basis, Zorn's lemma, and the well-ordering principle.

For two sets $A$ and $B$, there are several operations that are frequently used:

• $A\cup B$, the union, contains any element that is in $A$ or $B.$

• $A\cap B$, the intersection, contains any element that is in $A$ and $B.$

• $A^c$, the complement of $A$, contains all the elements not in $A.$

• $A\Delta B$, the symmetric difference of $A$ and $B$, contains the elements that are in exactly one of $A$ and $B.$

• $A\times B$, the Cartesian product of $A$ and $B$, contains the ordered pairs $(a,b)$ for $a\in A$ and $b\in B$.

• $2^{A}$, the power set of $A$, contains all the subsets of $A.$

An important quantity related to a set $A$ is the cardinality of a set $A$, denoted $|A|$, which counts how many elements the set $A$ contains. If this is finite, then it is just a number, but if it is infinite, there are some distinctions worth making.

There are countably infinite sets as well as uncountably infinite. Here, a countable set is one that can be put in bijection with the positive integers. An uncountable set is, in a precise sense, larger than a countable set, even though both have infinitely many elements. In fact, the power set of a countable set is an uncountable set.

The question of whether there is a size of infinite in between countable and the cardinality of $\mathbb{R}$ is known as the continuum hypothesis, and remains open, although it has been proven that this problem is independent of ZFC. This means that it can be neither proven nor disproven with the axioms of ZFC set theory.

Sets are also of fundamental importance in a branch of topology known as point set topology. A topology is a collection of sets called open sets. The complements of these open sets are called closed sets.

A topology $\tau$ in a space $X$ is a collection of sets satisfying the following properties:

• $\tau$ contains the empty set and $X.$
• The arbitrary union of sets in $\tau$ is in $\tau.$
• The finite intersection of sets in $\tau$ is in $\tau.$

Looking at these open sets can provide a great deal of insight into the structure of a space, since they give rise to the notions like compactness, which are extremely important in analysis.