2021/4/21
Zermelo-Fraenkel Axioms
INTRODUCTION
EXPOSITION
World of
Sets
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

Zermelo-Fraenkel Axioms
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 

Axiom of Extensionality (AEx)
Formal
Ground
∀x∀y(∀z(z ∈ x ⟷ z ∈ y) ⟶ x = y)
Axiom of
Extensionality
Axiom
Emptyof Set (AEm)
andofAxiom
Empty Set
∃x∀y¬(y ∈ x)
Frege's
Approach
and
Russell's of Specification (ASp)
Axiom-Schema
Paradox
Axiom-∀y1 ...∀yn (∀z∃y∀x(x ∈ y ⟷ x ∈ z ∧ φ(x; y1 , ..., yn )))
Schema of
Specification
Axiom of Pairing (APr)
More
Axioms
∀x∀y∃z∀u(u ∈ z ⟷ u = x ∨ u = y)
ZermeloFraenkel
Axiom
of Union (AUn)
Axioms
∀x∃y∀z(z ∈ y ⟷ ∃u(u ∈ x ∧ z ∈ u))
Axiom of Power Set (APw)
∀x∃y∀z(z ∈ y ⟷ ∀u(u ∈ z ⟶ u ∈ x))
Axiom of Infinity (AIn)
∃y(∃z(z ∈ y ∧ ∀t¬(t ∈ z)) ∧ ∀x(x ∈ y ⟶ ∃u(u ∈ y ∧ ∀v(v ∈ u ⟷
Axiom-Schema of Replacement (ARep)
∀y1 ...∀yn [∀x∀y∀y ′ (φ(x, y; y1 , ..., yn ) ∧ φ(x, y ′ ; y1 , ..., yn ) ⟶ y = y ′ )
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2021/4/21
Zermelo-Fraenkel Axioms
LOGIN JOIN
INTRODUCTION
⟶ ∀u∃v∀z(z
∈ v ⟷ ∃w(w ∈ u ∧ φ(w,
z; y
yn )))]
1 , ...,


Axiom of Choice (AC)
∀x∃f (f is a function ∧ ∀y(y ∈ x ∧ y =
 ∅ ⟶ f (y) ∈ y))
Axiom of Regularity (AReg)
∀x(∃y(y ∈ x) ⟶ ∃z(z ∈ x ∧ ∀u¬(u ∈ z ∧ u ∈ x))
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