پیشنیازها Naïve set theory History and evolution 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 Georg Cantor Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen Bertrand Russell Russell's Correspondence with Frege Ernst Friedrich Ferdinand Zermelo Investigations in the foundations of set theory I Set theory 1.∃𝐸 ∀𝑥(𝑥∉𝐸) ∃E ∀x(x∉E) forsome existsE forall x(x∉E)Axiom of Empty setaxiom✹ 2.∀𝐴 ∀𝐵 [∀𝑥(𝑥∈𝐴⇔𝑥∈𝐵) ⇒𝐴=𝐵] ∀A ∀B [∀x(x∈A⇔x∈B) ⇒A=B] forall A forall B [forall x(x∈A⇔x∈B) ⇒A=B]Axiom of Extensionalityaxiom✹ 3.∀𝐴 ∃𝐵 ∀𝑥(𝑥∈𝐵 ⇔ 𝑥∈𝐴 ∧ φ(𝑥)) ∀A ∃B ∀x(x∈B ⇔ x∈A ∧ φ(x)) forall A forsome existsB forall x(x∈B ⇔ x∈A ∧ φ(x))Axiom schema of separationaxiom✹ 4.∀𝑆 ∃𝑃 ∀𝑥(𝑥∈𝑃 ⇔ 𝑥⊂𝑆) ∀S ∃P ∀x(x∈P ⇔ x⊂S) forall S forsome existsP forall x(x∈P ⇔ x⊂S)Axiom of the power setaxiom✹ 5.∀𝐴 ∀𝐵 ∃𝐶 ∀𝑥(𝑥∈𝐶 ⇔ 𝑥=𝐴 ∨ 𝑥=𝐶) ∀A ∀B ∃C ∀x(x∈C ⇔ x=A ∨ x=C) forall A forall B forsome existsC forall x(x∈C ⇔ x=A ∨ x=C)Axiom of pairingaxiom✹ 6.∀𝑆 ∃𝑈 ∀𝑥[𝑥∈𝑈 ⇔ ∃𝐴(𝑥∈𝐴 ∧ 𝐴∈𝑆)] ∀S ∃U ∀x[x∈U ⇔ ∃A(x∈A ∧ A∈S)] forall S forsome existsU forall x[x∈U ⇔ forsome existsA(x∈A ∧ A∈S)]Axiom of the unionaxiom✹ 7.∀𝑆 [𝑆≠∅ ⇒ ∃𝑥(𝑥∈𝑆 ∧ 𝑥∩𝑆=∅)] ∀S [S≠∅ ⇒ ∃x(x∈S ∧ x∩S=∅)] forall S [S≠∅ ⇒ forsome existsx(x∈S ∧ x∩S=∅)]Axiom of Foundationaxiom✹ 8.∀𝐴 ∃𝐵 ∀𝑦 [𝑦∈𝐵 ⇔ ∃𝑥 (𝑥∈𝐴 ∧ 𝑓(𝑥,𝑦))] ∀A ∃B ∀y [y∈B ⇔ ∃x (x∈A ∧ 𝑓(x,y))] forall A forsome existsB forall y [y∈B ⇔ forsome existsx (x∈A ∧ 𝑓(x,y))]Axiom of Replacementaxiom✹ 9.∃𝐼 (∅∈𝐼 ∧ ∀𝑥 (𝑥∈𝐼 ⇒ 𝑥∪{𝑥}∈𝐼)) ∃I (∅∈I ∧ ∀x (x∈I ⇒ x∪{x}∈I)) forsome existsI (∅∈I ∧ forall x (x∈I ⇒ x∪{x}∈I))Axiom of infinityaxiom✹ 10.∀𝑋 [∅∉𝑋 ⇒ ∃𝑓 :𝑋→⋃𝑋 ∀𝐴∈𝑋 (𝑓(𝐴)∈𝐴)] ∀X [∅∉X ⇒ ∃𝑓 :X→⋃X ∀A∈X (𝑓(A)∈A)] forall X [∅∉X ⇒ forsome exists𝑓 :X→⋃X forall A∈X (𝑓(A)∈A)] Axiom of choiceaxiom✹