View source for Notes:The foundations of Mathematics - Kunen

==Chapter I: Set Theory==
==Section I.1==
==Section I.2: The axioms==
{| class="wikitable" border="1"
|-
! #
! Axiom
! Definition
! Comment
|-
|0
! Existence
| {{M|1=\exists x(x=x)}}
|-
|1
! Extensionality
| {{M|1=\forall z(x\in x\leftrightarrow z\in y)\rightarrow x=y}}
|-
|2
! Foundation
| {{M|1=\exists y(y\in x)\rightarrow\exists y(y\in x\wedge\not\exists z(z\in x\wedge z\in y))}}
|-
|3
! Comprehension schema
| {{M|1=\exists y\forall x(x\in y\leftrightarrow x\in z\wedge\varphi(x))}}
| {{M|\varphi}} a formula, {{M|y}} not free
|-
|4
! Pairing
| {{M|1=\exists z(x\in z\wedge y\in z)}}
|-
|5
! Union
| {{M|1=\exists A\forall Y\forall x(x\in Y\wedge Y\in\mathcal{F}\rightarrow x\in A)}}
| Union of {{M|\mathcal{F} }}
|-
| 6
! Replacement schema
| {{M|1=\forall x\in A\exists!y\varphi(x,y)\rightarrow\exists B\forall x\in A\exists y\in B\varphi(x,y)}}
| For each formula, without {{M|B}} free
|-
| 7
! Infinity
| {{M|1=\exists x(\emptyset\in x\wedge\forall y\in x(S(y)\in x))}}
|-
| 8
! Power set
| {{M|1=\exists y\forall z(z\subseteq x\rightarrow z\in y)}}
|-
| 9
! Choice
| {{M|1=\emptyset\not\in F\wedge\forall x\in F\forall y\in F(x\neq y\rightarrow x\cap y=\emptyset)\rightarrow \exists C\forall x\in F(\text{Sing}(C\cap x))}}
|}
: '''Alec's note: ''' "axiom" 0 can be shown from the axiom of infinity.
===Theories===
{| class="wikitable" border="1"
|-
! rowspan="2" | Theory
! colspan="9" | Axioms
! rowspan="2" | Comment
|-
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
|-
! ZFC
| x
| x
| x
| x
| x
| x
| x
| x
| x
|-
! ZF
| x
| x
| x
| x
| x
| x
| x
| x
|
|-
! ZC
| x
| x
| x
| x
| x
| 
| x
| x
| x
|-
! Z
| x
| x
| x
| x
| x
| 
| x
| x
| 
|-
! Z<sup>-</sup>
| x
| 
| x
| x
| x
| 
| x
| x
| 
|-
! ZF<sup>-</sup>
| x
| 
| x
| x
| x
| x
| x
| x
|
|-
! ZC<sup>-</sup>
| x
| 
| x
| x
| x
| 
| x
| x
| x
|-
! ZFC<sup>-</sup>
| x
| 
| x
| x
| x
| x
| x
| x
| x
|}