1 Preuve topologique de la compacité du calcul proposi

X
F
XL=X∪ {¬,∧,∨}
A SA=c∈ {0,1}X| ∀ϕ∈A, [[ϕ]]c= 1
A A
[[ϕ]]cϕ(c)X
{0,1}X→ {0,1}c ϕ
SA=\
ϕ∈A
c∈ {0,1}X|ϕ(c)=1=\
ϕ∈A
ϕ−1({1})
ϕ n ∈Nx1,· · · xn∈X
ϕ V ={x1,· · · xn}
ϕ:{0,1}X→ {0,1} {0,1}
{∅,{0},{1},{0,1}} {0,1}X
∅ {0,1}ϕ
ε= 0 1 ϕ−1({ε})
{0,1}Xϕ
x /∈V c x
[[ϕ]]c+[x=1] = [[ϕ]]c+[x=0] = [[ϕ]]c
ϕ−1({ε}) = c+c0∈ {0,1}X
c∈ {0,1}V,[[ϕ]]c=ε c0∈ {0,1}X\V
Sϕ+{0,1}X\V
{0,1}X
ϕ
SA=\
ϕ∈A
ϕ−1({1})⊂ {0,1}X
ϕ{1} {0}ϕ−1({1})
{0,1}X
{0,1}X
A SA=∅
F⊂A SF=Tϕ∈Fϕ−1({1}) = ∅
X