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 $X$ is also an element of $Y$ and every element of $Y$ is also an element of $X$ 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 $P(x)$ of x, given a set $A$ there is a set $B$ such that $x\in B\iff x\in A\text{ and }p(x)$. 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 $A$ and a property $P$ 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 $A$ and $B$ there is a set $C$ 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 $S$ there exists a set $U$ 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		$A$ is a subset of $B$ if and only if every element of $A$ belongs to $B$, 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 $S$ there eixsts a set $\mathcal{P}$ such that $X\in\mathcal{P}\iff X\subset S$ (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	$$\exists S(\emptyset \in S\wedge \forall x(x\in S\rightarrow x\cup\{x\}\in S))$$