ZFC Set Theory

Axioms

Axiom of Extensionality

$\forall A,\forall B,(\forall C,(C\in A\iff C\in B)\Rightarrow A=B)$

Two sets are equal if they have the same elements

Axiom of Regularity

$\forall A,(A\ne \varnothing \Rightarrow \exists B,(B\in A\wedge B\cap A=\varnothing ))$

Every non-empty set $A$ contains a member of $B$ such that $A$ and $B$ are disjoint sets

This implies, $\nexists A,A\in A$

Axiom schema of Specification

$\forall \varphi ,\forall A,\exists B,\forall x,(x\in B\iff (x\in A\wedge \varphi (A)))$ where $\varphi$ is a Propositional Function

Every subset of a set, defined by a Propositional Function is itself a set

Axiom of Pairing

$\forall A,\forall B,\exists C,(A\in C\wedge B\in C)$

If $A,B$ are sets, then $\exists$ a set which contains $A,B$ as elements

Axiom of Existence

$\exists A,\forall x,\neg (x\in A)$

There is a set such that no element is a member of it

Axiom of Union

$\forall A,\exists B,B=\bigcup A$ where $A$ is a Family of Sets

The union over the elements of a set exists

Axiom Schema of Replacement

$\forall A,((\forall x\in A,\exists !y,x\mathcal{R}y)\Rightarrow \exists B,\forall y,(y\in B\iff \exists x\in A,x\mathcal{R}y))$ where $\mathcal{R}$ is a relation on $A$

The image of any set under definable function is also a set

Axiom of Infinity

$\exists A,(\varnothing \in A\wedge \forall x\in A,((x\cup \{x\})\in A))$

Guarantees the existence of at least one infinite set

Axiom of Power Set

$\forall A,\exists P,\forall B,(B\subset A\Rightarrow B\in P)$

For every set $A$, there is a set $2^A$ consisting precisely of the subsets of $A$

Axiom of Choice

$\forall S,\left[\neg (\varnothing \in S)\Rightarrow \left(\exists f:S\to \bigcup _{A\in S}A,\forall A\in S,(f(A)\in A)\right)\right]$ where $S$ is a Family of Sets, and $f:\{A_i\}\to X$ is a Choice Function

Given any collection of sets, each containing at least one element, it is possible to construct a new set by the Choice Function that chooses arbitrary one element from each set.

Equivalents

Hausdorff Maximal Principle

Definition

Every Partially Ordered Set has a maximal chain

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Zorn's Lemma

Definition

Every non-empty Partially Ordered Set in which every chain has an upper bound contains at least one Maximum

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Well-Ordering Theorem

Definition

Every set can be well-order

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