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