Difference between revisions of "Notes:The foundations of Mathematics - Kunen"

Latest revision as of 02:38, 31 July 2016

#	Axiom	Definition	Comment
0	Existence	[ilmath]\exists x(x=x)[/ilmath]
1	Extensionality	[ilmath]\forall z(x\in x\leftrightarrow z\in y)\rightarrow x=y[/ilmath]
2	Foundation	[ilmath]\exists y(y\in x)\rightarrow\exists y(y\in x\wedge\not\exists z(z\in x\wedge z\in y))[/ilmath]
3	Comprehension schema	[ilmath]\exists y\forall x(x\in y\leftrightarrow x\in z\wedge\varphi(x))[/ilmath]	[ilmath]\varphi[/ilmath] a formula, [ilmath]y[/ilmath] not free
4	Pairing	[ilmath]\exists z(x\in z\wedge y\in z)[/ilmath]
5	Union	[ilmath]\exists A\forall Y\forall x(x\in Y\wedge Y\in\mathcal{F}\rightarrow x\in A)[/ilmath]	Union of [ilmath]\mathcal{F} [/ilmath]
6	Replacement schema	[ilmath]\forall x\in A\exists!y\varphi(x,y)\rightarrow\exists B\forall x\in A\exists y\in B\varphi(x,y)[/ilmath]	For each formula, without [ilmath]B[/ilmath] free
7	Infinity	[ilmath]\exists x(\emptyset\in x\wedge\forall y\in x(S(y)\in x))[/ilmath]
8	Power set	[ilmath]\exists y\forall z(z\subseteq x\rightarrow z\in y)[/ilmath]
9	Choice	[ilmath]\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))[/ilmath]

Alec's note: "axiom" 0 can be shown from the axiom of infinity.