corrigé de la feuille 5

T T T
T
Mn
∀x1, . . . , xn
_
1≤i<j≤n
xi=xj
T T
T T
T
T
0 1 ∀x x2+ 1 6= 0 R
C
M N M N
N
L={=} L M ={0} N =NM
N M N
N M
M N M
N M N M
N L <
L M = (Q∩]− ∞,0], <)N= (Q∩]− ∞,1], <)<R
M N
0M ∀y(y < x ∨y=x)
N
T
n∈N∗En
∃x1, . . . , xn
^
1≤i<j≤n
xi6=xj
.
E {En:n∈N∗} E
T T
ET
TN= (N,0,1,+) T
(N,0,1,+) a

L+{0,1,+,{cm}m≥2, d}cmd
T+= ((N, m)m≥2)∪ {∃x(x+. . . +x
| {z }
m
=d)|m∈N∗}∪{d6= 0}.
T+
T+L N
L={Pi:i < ω}PiT Pi
PiT κ κ > ℵ0
T
T κ κ ℵ0
TM1M2M1
M2M1M2κ
κℵ0κ κ ℵ0
ℵ0
T κ
Tℵ0Nℵ0
Tℵ0
L
L={Ei|i < ω}
i < ω EiE0
Ei+1 Ei
Ei
∀x Ei(x, x)
∀x, y (Ei(x, y)→Ei(y, x))
∀x, y, z (Ei(x, y)∧Ei(y, z)→Ei(x, z))
Ei
E0
∀x, y E0(x, y)
i∈N
∀x, y (Ei+1(x, y)→Ei(x, y))
∀x∃y, z (Ei(x, y)∧Ei(x, z)∧ ¬Ei+1(y, z))
∀x1, x2, x3³V1≤i,j≤3Ei(xi, xj)→W1≤i6=j≤3Ei+1(xi, xj)´
Ei+1 EiEi
Ei+1 EiEi+1
M0= ({f∈2ω|i < ω j ≥i, f (i) = f(j).};Ei(x1, x2) (i < ω)) ,
i∈ω(σ1, σ2)∈EM0
iσ1di=σ2di

T
MT σ ∈M θ ∈M
M |=Ei(σ, θ)i < ω T
T
MTM
cmM
Eii < ω
am,k m∈M k < ω
m∈M i, k, k0< ω k 6=k0
Ei(am,k, cm)∧(am,k 6=am,k0).
T(M+)
M N +
N+N M
m M N Eim
i < ω N=M1M1
M2M∞=SMi
M
M N N
k∈N(a1, . . . , ak) (b1, . . . , bk)kM N
(k1, k2) 1 ≤k1, k2≤k i < ω
M |=Ei(ak1, ak2)N |=Ei(bk1, bk2)
ak1=ak2bk1=bk2
α∈M β ∈N(a1, . . . , ak, α) (b1, . . . , bk, β)
k(a1, . . . , ak)
(b1, . . . , bk)α∈M ai
bia∼b Ei(a, b)i < ω I ={i:ai∼α}
INβ∼
bii∈I biβ
I j ∈ {1, . . . , k}ij¬(ajEijα)
β
k
T

M N (a1, . . . , ak) (b1, . . . , bk)
φ(x1, . . . , xk)
(M |=φ(a1, . . . , ak)) ⇔(N |=φ(b1, . . . , bk)) .
φ
φ¬ψ ψ
φ∃xψ(x1, . . . , xn, x)ψ
φM |=φ(a1, . . . , ak)α∈MM |=
φ(a1, . . . , ak, α)β∈N(a1, . . . , ak, α) (b1, . . . , bk, β)
M |=ψ(a1, . . . , ak, α)⇔ N |=ψ(b1, . . . , bk, β).
φ φ
(M |=φ)⇔(N |=φ).
∃xφ(x)φ φ
T
T T
M N
(mk) (nk)M N
(ak)∈MN(bk)∈NN
∀k mk∈ {a1, . . . , a2k}
∀k nk∈ {b1, . . . , b2k+1}
∀k(a1, . . . , ak) (b1, . . . , bk)
M={ak}N={bk}
f:M→N f(ak) = bk
φ(x1, . . . , xk)i1, . . . , ik
M |=φ(ai1, . . . , aik)⇔ N |=φ(bi1, . . . , bik).
fM N
T φ(x1, . . . , xl) (l∈N∗)
lLψ(x1, . . . , xl)
T`¡∀x1. . . xlψ(x1, . . . , xl)↔φ(x1, . . . , xl))
φ(x1, . . . , xl)
l≥1lL
Γ(x1, . . . , xl) = {ψ(x1, . . . , xl): ψ T ` ∀x1, . . . , xlφ(x1, . . . , xl)→ψ(x1, . . . , xl)}.
l d1, . . . , dlT∪Γ(d1, . . . , dl)`φ(d1, . . . , dl)
ψ1(d), . . . , ψm(d)∈Γ(d)
T∪ {ψ1(d), . . . , ψm(d)} ` φ(d).

T` ∀x
n
^
i=1
ψi(x)↔φ(x).
T∪Γ(d)`φ(d)
MT∪Γ(d)∪ {¬φ(d)}
Σ(d)ψ(d)M |=ψ(d)T∪Σ(d)∪ {φ(d)}
ψ1(d), . . . , ψn(d)
T` ∀xÃn
^
i=1
ψi(x)→ ¬φ(x)!.
T` ∀xÃφ(x)→
n
_
i=1
¬ψi(x)!.
Wn
i=1 ¬ψi(d)∈Γ(d)M
Γ(d)
NT∪Σ(d)∪ {ϕ(d}
(a1, . . . , al)=(dM
1, . . . , dM
l) (b1, . . . , bl) = (dN
1, . . . , dN
l) (a1, . . . , al) (b1, . . . , bl)
Σ(d)
(a1, . . . , al) (b1, . . . , bl)
T ω