Notation t(x)
means that term t
does not contain any different
from x
variable. Then t(a)
denote the value of t(x) when the
value of x is a.
For a tuple x,
the sense of notations t(x) and
t(a) is similar.
Φ(x,y)
denote that formula Φ
does not contain free variables which are not in variable
sequences x or
y.
Let I
be an indiscernible sequence in M
without end points. The sequence is
called divided (in M)
if for any terms
t1(x) and
t2(x),
either there is a natural number n such that
t2(a)<n
for any a
from I,
or for any a<b
from I,
t1(a)<t2(b).
The definition was proposed by S.M.Dudakov (see [6]).
The structure (M,I)
is called divided if I
is divided in M.
Theorem 12 (Dudakov, [6])
For any unary f, there are
M and
I such that
M is
first-order equivalent to
< N, <, 0, 1,
+, {[/n] | n=1,2,…}, f >
and
I is divided in
M.
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