ZFC Axioms for Set

Zermelo-Fraenkel set theory, usually ZF or ZFC, is a first-order theory with equality in the language ${\mathcal{L}}_{\in }$having just the one non-logical binary relation $\in$, for membership. The idea is that in a model of set theory the sets themselves are the individual members of the model.

$(V,\in )\text{with}V\text{a set in the meta universe}$

Axiom Schema of Comprehension: for $P$ a property, a set $Y$ is defined as $Y:=\{x|P(x)\},$ this is false as if it were true we would run into Russell’s paradox(Let $P$ be $x\notin x$).

We want something works as empty set, intuitively $\exists x\forall y(y\notin x)$, but this definition won’t give us unique empty set, thus implies following axioms.

Axiom of Extensionality

Two sets are equal (are the same set) if they have the same elements $\forall u(u\in x⟺u\in y)\to x=y$

To define function $f:A\to B$, $f\subset A\times B$, $(a,b)\in f⟺f(a)=b$ we need definition of ordered pairs, thus having following axiom.

Axiom of Pairing

For any two sets $x$ and $y$ there is a set whose members are exactly $x$ and $y$, $\forall x,y\exists z\forall t(t\in z⟷t=x\vee t=y)$ We write $z=\{x,y\}$. If $x=y$ then we write $z=\{x\}$.

Union Axioms

For every set $x$ there exists a set $y$ such that $\forall x\exists y\forall t(t\in y⟷\exists z(t\in z\wedge z\in x)).$ We write $y=\bigcup x.$

For example: $x=\{\{a,b\},\{c\}\}$ then $y=\{a,b,c\}$.

Power set Axiom

For every set $x$ there exists a set containing all subsets of $x$: $\forall x\exists y\forall z(z\in y⟷z\subseteq x).$ We write $y=\mathcal{P}(x).$

We call $\mathcal{P}(A)$ the power set of $A$. As an example, let us see what happens when we start with the empty set and take power sets over and over. Define $V_0=\varnothing ,$ $V_1=\mathcal{P}(V_0)=\{\varnothing \},$ $V_2=\mathcal{P}(V_1)=\{\varnothing ,\{\varnothing \}\},$ $V_3=\mathcal{P}(V_2)=\{\varnothing ,\{\varnothing \},\{\{\varnothing \}\},\{\varnothing ,\{\varnothing \}\}\},$ and, in general, $V_{n+1}=\mathcal{P}(V_n).$

Seperation Axiom Schema

$\forall z_1\cdots \forall z_n\forall y\exists z\forall x(x\in z⟺(x\in y\wedge \phi (x,z_1,\cdots ,z_n)))$ we write $z=\{x\in y\mid \phi (x,z_1,\cdots ,z_n)\}$

Remark

We get this separation axiom to avoid Rossel’s paradox. i.e. the set $z=\{x\mid x\notin x\}$. The reason that we successfully avoid the paradox through this axiom is we get a “legal” y as a set at the first place, and consider $x\in y$.

Axiom of Infinity

There exists an infinite set ( a particular type of infinite set): $\exists x(\varnothing \in x\wedge (\forall y\in x)y\cup \{y\}\in x)$

This allows us to define the natural numbers: $0:=\varnothing ,1:=\{0\},2:=\{0,1\},3:=\{0,1,2\}$ We define $\omega =\{0,1,2,3,4,\dots \}.$

Axiom of Replacement

If a class $F$ is a function, then for any $X$ there exists a set $Y=F(X)=\{F(x):x\in X\}$. For a formula $\phi (x,y,v_1,\dots ,v_k)$: $\forall x\forall y_1\forall y_2\left(\right(\phi (x,y_1,v_1,\dots ,v_k)\wedge \phi (x,y_2,v_1,\dots ,v_k))⟶y_1=y_2)$ implies $\forall t\exists u\forall y(y\in u⟷\exists x\in t\phi (x,y,v_1,\dots ,v_k)).$

Remark

Here we get a “well-defined function” through $\forall x\forall y_1\forall y_2\left(\right(\phi (x,y_1,v_1,\dots ,v_k)\wedge \phi (x,y_2,v_1,\dots ,v_k))⟶y_1=y_2),$ as the condition here mimicking for any “input x”, we get only one “output y”. And $\forall t$ here in the formula $\forall t\exists u\forall y(y\in u⟷\exists x\in t\phi (x,y,v_1,\dots ,v_k)).$ corresponds to $\forall X$.

Axiom of Foundation/ Regularity

Every non-empty set contains an $\in$-minimal element. $\forall x\ne \varnothing \exists y\in x(\forall z\in x(z\notin y)),$ Equivalently, we get $\forall x\ne \varnothing \exists y\in xy\cap x=\varnothing .$

Remark

This axiom avoids following conditions.

• No infinite descending chains. That is, no set $x=\{x_1,x_2,x_3,\dots \}$ such that $x_1\ni x_2\ni x_3\ni \dots .$
• No cycles. That is, no set $x=\{x_1,x_2,x_3,\dots ,x_n\}$ such that $x_1\in x_2\in x_3\in \cdots \in x_n\in x_1$.
• No set $x$ such that $x\in x$.

Redundancy in the Zermelo-Fraenkel Axiom System

In the standard formulation of Zermelo-Fraenkel set theory (ZF), certain axioms are strictly redundant, meaning they can be formally deduced from the remaining axioms. They are typically retained in textbooks for historical continuity and pedagogical clarity.

1. The Axiom of Empty Set The existence of the empty set can be deduced from the Axiom of Infinity and the Axiom Schema of Separation.

• Deduction: The Axiom of Infinity guarantees the existence of at least one set, say $X$. By applying the Axiom Schema of Separation with an always-false formula $\phi (x)\equiv (x\ne x)$, we can construct a subset $Y\subseteq X$ such that $Y=\{x\in X\mid x\ne x\}$. By the Axiom of Extensionality, this $Y$ is the unique empty set $\varnothing$.

2. The Axiom Schema of Separation (Specification) The Axiom Schema of Separation can be deduced entirely from the Axiom Schema of Replacement.

• Deduction: To construct a subset of $A$ containing only elements that satisfy a formula $\phi (x)$, we define a class function $F$ on $A$. If there is no element in $A$ satisfying $\phi (x)$, the subset is $\varnothing$. If there is at least one element $z_0\in A$ satisfying $\phi (z_0)$, we define $F(x)=x$ if $\phi (x)$ holds, and $F(x)=z_0$ if $\phi (x)$ fails. Applying Replacement to $A$ under $F$ yields exactly the separated subset $\{x\in A\mid \phi (x)\}$.

3. The Axiom of Pairing The Axiom of Pairing can be deduced from the Axiom of Power Set and the Axiom Schema of Replacement (assuming the existence of the empty set).

• Deduction: We need a “template” set of size 2. Starting with the empty set $\varnothing$, we apply the Power Set operation twice: $V_1=\mathcal{P}(\varnothing )=\{\varnothing \}$ $V_2=\mathcal{P}(V_1)=\{\varnothing ,\{\varnothing \}\}$ $V_2$ is a valid set containing exactly two distinct elements. To form the pair $\{a,b\}$ for any given sets $a$ and $b$, we apply the Axiom Schema of Replacement to $V_2$ with the mapping $F(\varnothing )=a$ and $F(\{\varnothing \})=b$.

Minimalist ZF Core: Consequently, an equivalent and minimalist axiomatization of ZF only strictly requires: Extensionality, Infinity, Union, Power Set, Replacement, and Regularity.