We prove here the following theorems: A. If k is a denumerable Hilbertian field then for almost all
(
σ
1
,
…
,
σ
e
)
∈
G
(
k
s
/
k
)
e
({\sigma _1}, \ldots ,{\sigma _e}) \in \mathcal {G}{({k_s}/k)^e}
the fixed field of
{
σ
1
,
…
,
σ
e
}
,
k
s
(
σ
1
,
…
,
σ
e
)
\{ {\sigma _1}, \ldots ,{\sigma _e}\} ,{k_s}({\sigma _1}, \ldots ,{\sigma _e})
, has the following property: For any non-void absolutely irreducible variety V defined over
k
s
(
σ
1
,
…
,
σ
e
)
{k_s}({\sigma _1}, \ldots ,{\sigma _e})
the set of points of V rational over K is not empty. B. If E is an elementary statement about fields then the measure of the set of
σ
∈
G
(
Q
~
/
Q
)
\sigma \in \mathcal {G}(\tilde Q/Q)
(Q is the field of rational numbers) for which E holds in
Q
~
(
σ
)
\tilde Q(\sigma )
is equal to the Dirichlet density of the set of primes p for which E holds in the field
F
p
{F_p}
of p elements.