Axiome du choix - Normalesup.org

=∅=P(∅) = {∅} =P( ) = {∅,{∅}}, . . . N
n
N N
<+×
Z Q R R R

A
A f :A\{∅} → ∪X∈AX X ∈A\{∅}
f(X)∈X
A
A

R3
ω
E
E
P(E)
E
N
N
E
P(E)

(E, ≤)C∈ P(E)
C C
≤(x, y)∈C2x≤y y ≤x
(E, ≤)
E E
(E, ≤)
x∈E∀y∈E, (x≤y)⇒(x=y)
E E
E×E
(x, y)∈E2xRy (x, y)∈R
E
R∀x∈E, ∃y∈E, xRy
(xn)n∈NE∀n∈N, xnRxn+1
ω
(AC)⇒(ACD)⇒(ACω)
FP(E)
(xn)n∈N
x0=F(E)∀n∈N, xn+1 =F({x∈E, xnRx})
Qn∈NXnS
f Dom(f)f{1, . . . , m}m∈N
∀n∈Dom(f), f(n)∈Xn
S fRg Dom(f)(Dom(g)g|Dom(f)=f
(S, R) (fm)m∈N
∀m∈N, fmRfm+1
f=∪m∈Nfmfm
fNQn∈NXn

(Xi)i∈IX
FXF
∩F∈F F
EF
EF∗
C E ∪c∈Cc C
F∗
F∗F∗
i∈I πiX Xi
Qi∈I∩A∈F∗πi(A)
x∈Qi∈I∩A∈F∗πi(A)UQi∈IXix
U∈ F∗
B O O
¯
B∪O6=∅B∪O6=∅
x∈ ∩F∈F F