Overview

Set theory begins with two primitive notions of sets and membership. Other axioms are defined relative to these concepts.

Sets are often denoted using roster notation in which members are specified explicitly in a comma-delimited list surrounded by curly braces. Alternatively, abstraction (or set-builder notation) defines sets using an entrance requirement. Examples of the set of prime numbers less than $10$:

• Roster notation: $\{2,3,5,7\}$
• Set-builder notation: $\{x\mid x<10\wedge x\text{ is prime}\}$

Extensionality

If two sets have exactly the same members, then they are equal: $\forall A,\forall B,(\forall x,x\in A\iff x\in B)\Rightarrow A=B$

Empty Set Axiom

There exists a set having no members: $\exists B,\forall x,x\notin B$

Pairing Axiom

For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$: $\forall u,\forall v,\exists B,\forall x,(x\in B\iff x=u\vee x=v)$

Union Axiom

Preliminary Form

For any sets $a$ and $b$, there exists a set whose members are those sets belonging either to $a$ or to $b$ (or both): $\forall a,\forall b,\exists B,\forall x,(x\in B\iff x\in a\vee x\in b)$

General Form

For any set $A$, there exists a set $B$ whose elements are exactly the members of the members of $A$: $\forall A,\exists B,\forall x,x\in B\iff (\exists b\in B,x\in b)$

Power Set Axiom

For any set $a$, there is a set whose members are exactly the subsets of $a$: $\forall a,\exists B,\forall x,(x\in B\iff x\subseteq a)$

Subset Axioms

For each formula $\_\_\_$ not containing $B$, the following is an axiom: $\forall t_1,\cdots ,\forall t_k,\forall c,\exists B,\forall x,(x\in B\iff x\in c\wedge \_\_\_)$

Axiom of Choice

This axiom assumes the existence of some choice function capable of selecting some element from a nonempty set. Note this axiom is controversial because it is non-constructive: there is no procedure we can follow to decide which element was chosen.

Relation Form

For any relation $R$ there exists a function $F\subseteq R$ with $\mathrm{dom}F=\mathrm{dom}R$.

Infinite Cartesian Product Form

For any set $I$ and function $H$ with domain $I$, if $H(i)\ne \varnothing$ for all $i\in I$, then ${\text{bigtimes}}_{i\in I}H(i)\ne \varnothing$.

Infinity Axiom

There exists an inductive set: $\exists A,[\varnothing \in A\wedge (\forall a\in A,a^{+}\in A)]$

Bibliography

• “Axiom of Choice,” in Wikipedia, July 8, 2024, https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262.
• Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
• “Russell’s Paradox,” in Wikipedia, April 18, 2024, https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437.
• Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).