El axioma de Sustitución

α a α ∈ a
∀x ((x ∈ a) → (x ′ ∈ a)) α = ∅
α = ω
U
α ∅

a0 = ω ω0 = ∪a0 = ω
a1 = {ω0, ω ′ 0, ω ′′
0 . . .} w1 = ∪a1
n ∈ ω an
an ′ ∪an
an n ∈ ω
I i ∈ I ai
a ai ∈ a i ∈ I
I = ω an = α (n) (n)
n I = ω a0 = ω
an ′ = {∪an, (∪an) ′ , (∪an) ′′ . . .}
a
ϕ(x, y) ∀x ((x ∈ a) → ∃!y ϕ(x, y)
U
x a
F (x) ϕ
{F (x) : x ∈ a}
ϕ
x y t ϕ
∀z {[ ∀x ∀y ∀t (x ∈ z ∧ ϕ(x, y) ∧ ϕ(x, y|t)) → y = t ] →
∃u [∀y (∃x (x ∈ z ∧ ϕ(x, y))) ↔ (y ∈ u)]}.
∀z[∀x((x ∈ z) → (∃!y(ϕ(x, y))] → ∃u [∀y (∃x (x ∈ z ∧ ϕ(x, y))) ↔ (y ∈ u)].
F U a
f a
b = {y : ∃x (x ∈ a ∧ y = F (x))}
f = {r ∈ a × b : ∃x ∃y (r = 〈x, y〉 ∧ y = F (x))}.
f a b
f = F |a
ϕ(x, y|t)

ψ
ϕ(x, y) (x = y ∧ ψ(y))
∀x∀v ((ϕ(x, y) ∧ ϕ(x, y|v)) → (y = x = v))
∀z ∃u ∀y [(y ∈ u) ↔ (∃x ((x ∈ z) ∧ (y = x) ∧ ψ(y)))],
u, v
{u, v}
∅ = P(∅)
x ∈ P(∅) ↔ x = ∅
x ∈ P(P(∅)) ↔ (x = ∅ ∨ x = P(∅))
ϕ(z, y, u, v) (z = ∅ ∧ y = u) ∨ (z = P(∅) ∧ y = v)
z ∈ P(P(∅)) ϕ(y) ∧ ϕ(y|t) y = t ϕ
P(P(∅))
∀u∀v[∃x(y ∈ x ↔ (z ∈ P(P(∅)))) ∧ ϕ(y)],
∀u∀v[∃x(y ∈ x ↔ (z ∈ P(P(∅)))) ∧ (z = ∅ ∧ y = u) ∨ (z = P(∅) ∧ y = v)]
∀u∀v[∃x(y ∈ x ↔ (y = u ∨ y = v)]

a b f : a → b
x y a x ≤ y
f(x) ≤ f(y) x ≤ y f(x) ≤ f(y) f
a b a ∼ b a
b
≤ a b
x, y a x ≤ y a
f(x) ≤ f(y) b
a s(a)
a
f : a → s(a) f(x) = ax x ∈ a
a ∼ s(a)
a b f : a → b
f
a = {x, y}
x ≤ y ↔ x = y b = {s, t} s < t
f : a → b f(x) = t f(y) = s
f a Im(f) ⊆ b
f f
x ∈ a f ax ax
b f(x)
ā = ¯ b a ∼ b
ā = ¯ b
a ∼ b b ∼ a

α β f : α → β
α = β f
f α β a = {ζ ∈ α : f(ζ) = ζ}
a = ∅ α γ
a ζ ∈ γ ζ /∈ a ζ = f(ζ) f
f(ζ) ∈ f(γ) ζ ∈ f(γ)
γ ⊆ f(γ)
γ ⊆ f(γ) γ = f(γ) γ ∈ f(γ)
f(γ) ∈ β β γ ∈ β
f δ ∈ α f(δ) = γ
δ ∈ a δ /∈ a
δ /∈ a δ = f(δ) = γ ∈ a
δ ∈ a γ ⊆ δ f(γ) ⊆ f(δ) = γ f(γ) ⊆ γ
γ ⊆ f(γ) γ = f(γ) γ ∈ a
a f
f ∀x(x ∈ α ↔ x ∈ β)
α = β
a γ
a ∼ γ
ϕ(x, y) (Ord(y) ∧ ax ∼ y)
b = {x ∈ a : ∃βϕ(x, β)}.
x ∈ a ϕ(x, y) ∧ ϕ(x, v) y v
ϕ a
c
y ∈ c ↔ Ord(y) ∧ ∃x(x ∈ a ∧ ϕ(x, y).
c α ∈ c x ∈ a f
α ax β ∈ α β = αβ ∼ a f(β)
β ∈ c c
b a x b y ∈ a
y < x α f ax α y ∈ ax

f(y) ∈ α f(y)
ay ∼ f(y) y ∈ b
b c
F : b → c F (x) ax
F x y b x < y
g ay F (y) g ax
ax g(x) F F (x) = g(x) ∈ F (y) x < y
F (x) < F (y) α, β c β ∈ α x, y ∈ b
α = F (x) β = F (y) x ≤ y F (x) ≤ F (y)
β ∈ α ⊆ β y < x F
F b ∼ c
b ∼ α x b = ax ax ∼ α
x ∈ b = ax b
b = a
γ = c
u
n ∈ ω f
u ωn
ϕ(x, y)
F : U → U
F (x) = y ↔ ϕ(x, y) Ord
T : Ord → U α T (α) = F (T |α)
T |α = {〈β, T (β)〉 : β ∈ α} α
T |α
α T |α : α → {T (β) : β ∈ α}
F
α f α
dom(f) = α f(β) = F (f|β) β ∈ α
β ∈ α f α f|β β
γ ∈ β (f|β)|γ = f|γ
α α
f g α dom(f) = dom(g) =
α α
γ γ

β ∈ γ f(β) = g(β) f|γ = g|γ f g α
γ ∈ α f(γ) = F (f|γ) = F (g|γ) = g(γ)
f(γ) = g(γ)
F (f) x = α f α
T (x) =
∅
α f T (α) = F (T |α)
β ∈ α f|β β
f(β) = F (f|β) = T (β) T |α = f|α = f
T (α) = F (f) = F (T |α)
α α
α α
β ∈ α β
T (β) = F (T |β) β ∈ α f = T |α
dom(f) = α β ∈ α f(β) = F (T |β) = F (f|β)
f α ϕ(x) ↔ Ord(x) ∧ ∃f x
∀α(∀β(β ∈ α ∧ ϕ(β)) → ϕ(α))
∀αϕ(α) α α
T α T (α) = F (T |α)
a0 F1 F2
α
⎧
⎨ a0 α = ∅
T (α) = F1(T (β)) α = β
⎩
′
F2(T |α) α
⎧
a0 x = ∅,
⎪⎨
F1(f(β)) x = β
F (x) =
⎪⎩
′ f β ∈ dom(f),
F2(x) x
∅ .
T
T (α) = F (T |α) T (0) = F (T |0) = a0 T (β ′ ) = F (T |β ′) = F1(T |β ′(β)) =
F1(T (β)) α T (α) = F (T |α) = F2(T |α)

≤ ω
a T
T (∅) = a
T (n ′ ) = ∪T (n) n ∈ ω
T (ω) =
n∈ω T (n)
a0 = a F1(x) = ∪x F2(x) = ∪img(x)
T T (0) = a
T (n ′ ) = F1(T (n)) = ∪T (n) T (ω) = F2(T |ω) = ∪{T (n) : n ∈ ω}
T (ω) a
a
a ⊆ T (ω)
T rans(T (ω))
a ⊆ b T rans(b) T (ω) ⊆ b
a = T (0) ⊆ ∪{T (n) : n ∈ ω}
x ∈ T (ω) n0 ∈ ω x ∈ T (n0) x ⊆ ∪T (n0) = T (n ′ 0) ⊆
T (ω) T (ω)
T (0) = a ⊆ b T (n) ⊆ b T (n ′ ) =
∪T (n) ⊆ ∪b = b b
n ∈ ω T (n) ⊆ b ω
T (ω) =
n∈ω T (n) ⊆ b
0 0 ′ 1 1 ′ = 0 ′′ 2
(a, r) (b, s) a∩b = ∅
a b a ⊔ b
(a ∪ b, r ∪ s ∪ (a × b)).
x ≤ y a ⊔ b
x y a 〈x, y〉 ∈ r
x y b 〈x, y〉 ∈ s

x ∈ a y ∈ b
a b a∩b = ∅
a ⊔ b a a ⊔ b
α x α × {x}
(ζ, x) ≤ (η, x) ζ ≤ η ζ η
α ζ ↦→ (ζ, x)
α α × {x} α β
α × {0} β × {1} α
β
α β α β
α ⊕ β α × {0} ⊔ β × {1}
a b a ∼ α
b ∼ β a ⊔ b ∼ α × {0} ⊔ β × {1} α ⊕ β
α β
α β γ
0 ⊕ α = α ⊕ 0 = α
α ⊕ (β ⊕ γ) = (α ⊕ β) ⊕ γ
α ⊕ 1 = α ′
α ⊕ β ′ = (α ⊕ β) ′
0 × {0} ⊔ α × {1} = ∅ ⊔ α × {1} ∼
α ∼ α × {0} ⊔ ∅ = α × {0} ⊔ ∅ × {0}
a b c
〈x, 〈y, z〉〉 ↦→ 〈〈x, y〉, z〉
a ⊔ (b ⊔ c) (a ⊔ b) ⊔ c
f : α ′ ↦→ (α×{0}) ⊔ (1×{1}) f(ζ) =
α ⊕ β ′ = α ⊕ (β ⊕ 1) = (α ⊕ β) ⊕ 1 = (α ⊕ β) ′
(ζ, 0) ζ ∈ α
(0, 1) ζ = α

f : 1 × {0} ⊔ ω × {1} ↦→ ω f((0, 0)) = 0 f((n, 1)) = n ′
1 ⊕ ω = ω
ω ⊕ 1 = ω ′ = ω
0 ⊕ ω = 1 ⊕ ω = ω 0 = 1
α β α ∈ β β \ α
β
β \ α
β ⊖ α β ⊖ β = 0
α β γ
α ∈ β β = α ⊕ (β ⊖ α)
β ∈ γ α ⊕ β ∈ α ⊕ γ
α ⊕ β = α ⊕ γ β = γ
f : β → α × {0} ⊔ (β \ α) × {1}
(ζ, 0) ζ ∈ α
f(ζ) =
(ζ, 1) ζ ∈ β \ α
β ∈ γ γ = β ⊕ (γ ⊖ β)
α ⊕ γ = (α ⊕ β) ⊕ (γ ⊖ β) γ ⊖ β = 0
α ⊕ β ∈ α ⊕ γ
α ⊕ γ = α ⊕ β β = γ
m n
m ⊕ n ∈ ω
m ⊕ n = n ⊕ m
m ∈ ω am = {n ∈ ω : m ⊕ n ∈ ω} am
0 ∈ am m ⊕ 0 = m ∈ ω n ∈ am
m ⊕ n ∈ ω m ⊕ n ′ = (m ⊕ n) ′ ∈ ω
m × {0} ⊔ n × {1} = n × {0} ⊔ m × {1}
m ⊕ n = n ⊕ m m ⊕ n n ⊕ m
m ⊕ n = n ⊕ m
ω × ω ↦→ ω m + 0 = m
m + n ′ = (m + n) ′
m ⊕ n

I i ∈ I (ai, ri)
ai ∩ aj = ∅ i = j
{ai}i∈I
ai = (
ai,
ri ∪
(ai × aj)).
i∈I
i∈I
i∈I
x ≤ y
i∈I ai
i

α β γ
0 ⊙ α = α ⊙ 0 = 0
1 ⊙ α = α ⊙ 1 = α
α ⊙ (β ⊕ γ) = (α ⊙ β) ⊕ (α ⊙ γ)
α > 0 β > 1 α ∈ α ⊙ β
α > 0 α ⊙ β = α ⊙ γ β = γ
α ⊙ β ′ = (α ⊙ β) ⊕ α
ζ∈β (α × {ζ}) = α × β
α × β
(γ, ζ) (δ, η) α × β (γ, ζ) < (δ, η)
ζ < η ζ = η γ < δ
m b
m ⊙ n ∈ ω
m ⊙ n = m × n
m ⊙ n = n ⊙ m
ω × ω ↦→ ω m · 0 = 0
m · n ′ = (m · n) + m
m ⊙ n
α β
α ⊕ β =
ξ∈β (α ⊕ ξ)
α ⊙ β =
ξ∈β (α ⊙ ξ)

α
Sα = {〈β, γ〉 : γ = α ⊕ β}
S(0) = α
S(β ′ ) = F (β) ′
S(β) =
ξ∈β S(ξ) β
α Gα : U → U x
⎧
α x = 0
⎪⎨
(x(β))
Gα(x) =
⎪⎩
′ x dom(x) = β ′
ξ∈β x(ξ) x dom(x) = β β ,
∅ .
Fα : Ord → Ord
a), b) c)
α
Pα = {〈β, γ〉 : γ = α ⊙ β}
P (0) = 0
P (β ′ ) = P (β) ⊕ β
P (β) =
ξ∈β P (ξ) β
α Hα : U → U x
⎧
⎪⎨ (x(β)) ⊕ α x dom(x) = β
Hα(x) =
⎪⎩
′ img(x) ⊂ Ord
ξ∈β x(ξ) x dom(x) = β β ,
∅ .
Gα : Ord → Ord
a), b) c)

α Eα : Ord → Ord
Eα(0) = 1
Eα(β ′ ) = Eα(β) ⊙ α
Eα(β) =
ξ∈β Eα(ξ) β
Eα(β) = α β
α, β, γ
α β ⊙ α γ = α β⊕γ
(α β ) γ = α β⊙γ
m, n ∈ ω m n ∈ ω
2 ω = ω