# Set theory axioms

This page is supposed to provide some discussion for the axioms (for example "there exists a set with no elements" doesn't really deserve its own page)

## List of axioms

The number column describes the order of introduction in the motivation for set theory axioms page, note that "R" denotes a result. Only "major" results are shown, they are covered in the motivation for set theory page, and "D" denotes "definition" - which is something that is natural to define at that point

Number Axiom Description Formal statement
1 Existence There exists a set with no elements
2 Extensionality (Equality) If every element of [ilmath]X[/ilmath] is also an element of [ilmath]Y[/ilmath] and every element of [ilmath]Y[/ilmath] is also an element of [ilmath]X[/ilmath] then
$\forall X\forall Y(\forall u(u\in X\leftrightarrow u\in Y)\rightarrow X=Y)$
R The empty set is unique can now be proved, and thus denoted $\emptyset$
3 Schema of Comprehension For a property [ilmath]P(x)[/ilmath] of x, given a set [ilmath]A[/ilmath] there is a set [ilmath]B[/ilmath] such that [ilmath]x\in B\iff x\in A\text{ and }p(x)[/ilmath].
A property may be $P(x):=x\in A=\phi(x,A)$ where $\phi$ is a formula
$\forall X\forall p\exists Y\forall u(u\in Y\leftrightarrow[u\in X\wedge\phi(u,p)])$
R For a set [ilmath]A[/ilmath] and a property [ilmath]P[/ilmath] the set known to exist by axiom 3 is unique, thus we may write $\{x\in A|P(x)\}$ to denote it unambiguously
4 Pair For any [ilmath]A[/ilmath] and [ilmath]B[/ilmath] there is a set [ilmath]C[/ilmath] such that $x\in C\iff x=A\text{ or }x=B$ $\forall A\forall B\exists C\forall x(x\in C\leftrightarrow x=A\vee x=B)$
R The set known to exist from axiom 4 is unique, we denote it by $\{A,B\}$ or $\{A\}$ if $A=B$ (at this point we "just write" this, we have no concept of cardinality yet)
R Ordered pair Kuratowski: $(a,b)=\{\{a\},\{a,b\}\}$
5 Union For any set [ilmath]S[/ilmath] there exists a set [ilmath]U[/ilmath] such that $x\in U\iff[x\in A\text{ for some }A\in S]$ $\forall X\exists U\forall s(s\in U\leftrightarrow \exists A(A\in X\wedge S\in A))$
R The union of a set S is unique, and thus denoted by $\cup S$
D [ilmath]A[/ilmath] is a subset of [ilmath]B[/ilmath] if and only if every element of [ilmath]A[/ilmath] belongs to [ilmath]B[/ilmath], that is $\forall x:x\in A\implies x\in B$ - we denote this $A\subset B$ $\forall u(u\in A\rightarrow u\in B)$$\iff \subset(A,B)\iff A\subset B$
6 Power set For any set [ilmath]S[/ilmath] there eixsts a set [ilmath]\mathcal{P} [/ilmath] such that [ilmath]X\in\mathcal{P}\iff X\subset S[/ilmath] (see Power set) $\forall X\exists\mathcal{P}\forall U(\forall s(s\in U\rightarrow s\in X)\leftrightarrow U\in\mathcal{P})$
7 Infinite set There exists an Inductive property $\exists S(\emptyset \in S\wedge \forall x(x\in S\rightarrow x\cup\{x\}\in S))$