Set Theory/Zermelo-Fraenkel (ZF) Axioms

Initial definitions

Before stating the axioms, we must first assume a knowledge of first order logic and its connectives and quantifiers. Our variables will simply represent sets in the universe of set theory, and so any letter—capital, lowercase, roman, Greek, etc.—will represent a set unless otherwise stated. We must also accept the basic property of membership so we can state that x is an element of y. For example, 1 is an element of $\{0,1\}$.

After defining appropriate axioms, we can freely use the equality symbol $=$ to mean that two sets are identical. See the axiom of extensionality below to see how equality is defined for sets.

Axioms of Set Theory

In order to prove some of the fundamental results of set theory, and to begin to define other branches of mathematics based on it, we need to start with some axioms that we can assume to be true. There are many possibilities for choices of axioms, but the most popular set of axioms is the Zermelo-Fraenkel system, or, more generally, Zermelo-Fraenkel with the Axiom of Choice.

We start by constructing the axiom system known as $Z_{0}$

We'll need a notion of equality between sets. The following axiom gives that.

Axiom of Extensionality: If $A$ and $B$ are sets such that $x\in A\iff x\in B$, then $A=B$.

We'll define a set that contains nothing:

Axiom of Empty Set: There is a set X such that for all $y$, $y\not \in X$.

Note that the empty set is written as $\emptyset$ or $\{\}$. Uniqueness of the empty set follows immediately from the axiom of Extensionality.

Now we need the ability to take a set, and divide up the elements based on whether or not they satisfy some property.

Axiom of Separation: If $A$ is a set and $P$ is a condition on elements of $A$, then $\{x:x\in A{\hbox{ and }}P(x)\}$ is a set.

The uniqueness of such a set follows again from the axiom of Extensionality. The condition $P$ must be described precisely using mathematical logic; a vague condition could lead to an ambiguous idea of what the set contains.

Axiom of separation is sometimes called schema of separation, since it comprises infinitely many axioms - one for each condition $P$. Notice that the restriction $x\in A$ in this axiom helps us to avoid Russell's paradox - in this paradox a "set" of the form $\{x:P(x)\}$, with $P(x)=x\not \in x$, was used.

The following three axioms are required to generate new sets from given sets.

Axiom of Pair: If $A$ and $B$ are sets, then there is a set containing exactly $A$ and $B$.

Axiom of Union: If $A$ is a set, then $\{x\mid x\in B{\hbox{ for some }}B\in A\}$ is a set. In other words, for any set A, there exists another set whose elements are precisely the elements of the elements of A.

Axiom of Power Set: If $A$ is a set, then there is a set ${\mathcal {P}}(A)$ so that $B\in {\mathcal {P}}(A)\iff$ each $b\in B$ is also in $A$.

These six axioms, taken together, allow the development of the operations that were discussed in the previous chapter.

Zermelo-Fraenkel

To derive certain fundamental results in various branches of mathematics, there is need for more axioms. While all mathematicians agree on the above six axioms, other axioms become controversial in various ways. The following axioms together with the above are called the Zermelo-Fraenkel axioms, often abbreviated ZF.

As we will see, there is no way to define an infinite set given our current axioms, so we require separate axioms for that.

Axiom of Infinity: An inductive set exists. An inductive set is a set I such that, $\emptyset \in I$, and $x\in I\Rightarrow x\cup \{x\}\in I$.

We shall see later why the following axiom is needed.

Axiom of Replacement: If $P(x,y)$ is a property such that for each x, there is a unique y such that $P(x,y)$ holds, then for every set A there is a set B such that for every $x\in A$ there is a $y\in B$ for which $P(x,y)$ holds.

Note that like the axiom of separation, the axiom of replacement is often called an axiom schema because there are infinitely many properties $P(x,y)$ to which such an axiom applies.

The following axiom is somewhat of a convention but various models of set theory have been defined without it, or even using axiom stating things close to the opposite.

Axiom of Regularity: For every non-empty set $x$ there is some $y\in x$ such that $x\cap y=\emptyset$.

The Axiom of Choice

The last Axiom in Zermelo-Fraenkel set theory is called the Axiom of Choice. It remains the most controversial axiom among mathematicians, and for that reason, when using ZF with the Axiom of Choice, it is often specified as ZFC.

Axiom of Choice: For every set S of nonempty disjoint sets, there exists a function f defined on S such that, for each set $x\in S$, $f(x)\in x$.