Let
fp(x)=y iff
(y|p x & ¬ ((py)|p x)).
Then
(y|p x) ⇔
(fp(x)≥y &
fp(y)=y).
So the first-order theories of
< N, <, +, |p>
and of
< N, <, +, fp>
are mutually interpreted.
For simplicity, suppose
p=2.
First, we point out that
< N, <, +, |2>
and
< N, <, ∈>
are mutually interpreted where
x∈y iff
(∃z)(∃u)
(y= z+x+u & z<x & (2x)|2u).
Second, we observe that
x<y
we may use only for x and
y such that
x∈ x and
y∈ y. Indeed,
u<v iff
(∃x)
(x∈ v&¬( x∈ u) &
(∀y)
(x<y →(y∈ u ↔ y∈ v))).
Thus, we will investigate the structure
< N, <, ∈>
where
< is defined on
{x | x∈ x} only.
Theorem 10
Let
M≡
< N, <, ∈ >.
Suppose
I is a subset of
{x∈ M | x∈ x}
such that
I is an indiscernible sequence in
M and
< is a dense order over
I. Then
Th(< N, <, ∈ >).
has no
(M,I)-Isolation Property.
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