Here is a rough overview of the basics of AbstractSetTheory, in particular ZFC and how we construct things like numbers and functions.

## What is 'abstract' about Abstract Set Theory?
We _never_ worry about, or ask about, what sets actually are. In the formal language of SetTheory, there is only one non-logical symbol: ∈. Abstract set theory is all about the relation ∈. We begin with a set of a priori assumptions about properties that any sensible notion of ∈ should satisfy. We call these axioms. There are a few different sets of axioms for Set Theory, but the most common is ZFC (Zermelo-Fraenkel with the AxiomOfChoice).

## Axioms
1. Extensionality: two sets are equal if and only if they have the same elements.
2. Regularity: every non-empty set contains an element \(y\) disjoint from \(x\).
3. Schema of Comprehension: Given a set \(X\) and a formula \(\phi(x)\), the set \(\{x\in X:\phi(x)\}\) exists.
4. Pairing: if \(x\) and \(y\) exist, then there is a set containing both \(x\) and \(y\).
5. Union: the union over the elements of a set exists. That is, if \(X = \{ x_1, x_2, \ldots \}\) then the set \(\bigcup X = x_1 \cup x_2 \cup \ldots\) exists.
6. Replacement: informally if \(f(x)\) is a function definable by a formula \(\phi\), \(A\) is a set, and \(f(x)\) exists for all \(x\in A\), then the set \(\{ f(x) : x\in A\}\) exists.
7. Infinity: there is a set containing \(\varnothing\) and closed under the successor operation \(x \rightarrow x \cup \{x\}\).
8. Power set: if \(x\) is a set, then so is the set of all subsets of \(x\), which we call the *power set* of \(x\) and denote it by \(\mathscr{P}(x)\).
9. Choice (AC): the cartesian product of a collection of nonempty sets is nonempty.
10. Or equivalent to AC: for every set \(X\) there is a binary relation on \(X\) which makes \(X\) well-ordered.

## Relations and functions
Given sets \(X\) and \(Y\), the **cartesian product** of \(X\) and \(Y\), denoted \(X\times Y\) is the set of all pairs \((x,y)\) such that \(x\in X\) and \(y\in Y\). That is: \(X\times Y=\{(x,y) : x\in X, y\in Y\}\). Similarly we define the cartesian product of many sets \(X_1\times X_2\times X_3\times\cdots\). The cartesian product of an infinite collection of sets \(\{ X_i : i\in I\}\) where \(I\) is an infinite *index set*, is more fiddly to define (essentially elements of the cartesian product of an infinite collection of sets are functions \(f:I\rightarrow\bigcup\{X_i:i\in I\}\) such that for all \(i\in I\) we have \(f(i)\in X_i\)).

A \(n\)-ary **relation** on a set \(X\) is a subset of \(X\times X\times\cdots\times X\) where the cartesian product is of \(n\) copies of \(X\).

A **function** \(f:X\rightarrow Y\) is a relation \(R(x,y)\) on \(X\times Y\), such that \(R\) associates an element of \(X\) to *at most one* element of \(Y\). That is, if \(R(x,y)\) and \(R(x,z)\) then necessarily \(y=z\). When \(R(x,y)\) defines a function, we use the notation \(f(x)=y\).

## Properties of functions
A function is **injective** if different inputs give different outputs. That is, if \(f(x)=z\) and \(f(y)=z\) then \(x=y\).

A function is **surjective** if its codomain is equal to its range. That is, if \(f:X\rightarrow Y\) then for all \(y\in Y\) there is \(x\in X\) with \(f(x)=y\).

A function is **bijective** if it is both injective and surjective.

## Ordinals
We can construct the natural numbers by identifying \(0\) with the empty set \(\varnothing\), and the proceeding to define the successor \(S(x)=x\cup\{x\}\). This way we get an *ordinal* for each natural number \(n\in\mathbb{N}\). These are the finite ordinals. We can take the union of all finite ordinals to get the first infinite ordinal, which we denote \(\omega\). We call these **limit ordinals**. Once we have \(\omega\), we can construct \(S(\omega\)\), which we denote \(\omega+1\), and then \(\omega+2\) and so on. Then we can take the union of all the \(\{\omega+n:n\in\omega\}\) to get \(\omega+\omega\), which we denote \(2\omega\). In this way, we can create bigger and bigger ordinals. Either using the successor operation or taking unions to produce limit ordinals. Ordinals are *well-ordered*. That is, a set of ordinals does not contain an infinite descending sequence.

## Cardinals
The **cardinality** of a set is the 'number' of elements in it. It can be infinite. Two sets have the same cardinality, that is, they are **equinumerous** if, and only if, there exists a *bijective* function between them.

The Schröder-Bernstein Theorem says that if there exists an injection \(f:X\rightarrow Y\) and an injection \(g:Y\rightarrow X\) then there exists a bijection \(h:X\rightarrow Y\).

We denote the cardinality of a set \(X\) by \(\|X\|\), and we identify \(\|X\|\) with the smallest ordinal \(\alpha\) such that \(X\) is equinumerous with \(\alpha\). The cardinal of the set of all finite ordinals is denoted \(\aleph_0\), (aleph null), and (assuming AC) is the smallest infinite ordinal.

If we assume the Axiom of Choice, or equivalently, the well-ordering principle, then every set is equinumerous with some ordinal, and so the cardinals are linearly ordered.

## Countability
A set is **countable** if either it is *finite* or else its cardinality is \(\aleph_0\). Equivalently, a set X is countable if it is the image of some function from \(\omega\). That is, if \(X=\{f(x):x\in\omega\}\).

The power set \(\mathscr{P}(X)\) of a set \(X\) is always of a larger cardinality than \(X\), as can be proved by Cantor's Diagonal Argument. The real numbers \(\mathbb{R}\) are at least as large as the power set \(\mathscr{P}(\mathbb{N})\) of the natural numbers, so are strictly larger (in cardinality terms), than \(\mathbb{N}\).