Information Technology Reference
In-Depth Information
(
iii
) We already have
V
(PS
3
)
|
=LO(
u
)from(
ii
). So it is sucient to prove
that
∀
y
(
y
ↆ
u
∧¬
(
y
=
∅
→∃
z
(
z
∈
y
∧
z
∩
y
=
∅
=1
,
5
)
))
i.e., for any
y
∈
V
(PS
3
)
if
y
ↆ
u
∧¬
(
y
=
∅
)
∈{
1
,
1
/
2
}
then
∃
z
(
z
∈
y
∧
z
∩
y
=
∅
. Now by definition and the fact that BQ
˕
hold in
V
(PS
3
)
for all
negation-free formulas
˕
,
)
∈{
1
,
1
/
2
}
=
t∈
dom(
y
)
y
ↆ
u
=
∀
t
(
t
∈
y
→
t
∈
u
)
(
y
(
t
)
⃒
t
∈
u
)
So,
y
ↆ
u
∈{
1
,
1
/
2
}
if and only if for any
t
∈
dom(
y
)if
y
(
t
)
=0then
t
∈
u
= 0, i.e., by Theorem 10 it can be concluded that
t
is
ʲ
-like for some
ʲ
∈
ʱ
. Again,
=
z∈
V
(PS
3
)
¬
(
y
=
∅
∃
z
(
z
∈
y
)
(
y
(
t
)
∧
z
=
t
)
=
)
t∈
dom(
y
)
Therefore
¬
(
y
=
∅
)
∈{
1
,
1
/
2
}
if and only if there exists
t
∈
dom(
y
) such that
y
(
t
)
∈{
1
,
1
/
2
}
.
Hence
y
ↆ
u
∧¬
(
y
=
∅
)
∈{
1
,
1
/
2
}
if and only if there exists
t
∈
dom(
y
)
such that
y
(
t
)
∈{
1
,
1
/
2
}
and for each
t
∈
dom(
y
)if
y
(
t
)
∈{
1
,
1
/
2
}
then
t
is
ʲ
-like
for some
ʲ
ʱ
.
Let us now find the value of
∈
∃
z
(
z
∈
y
∧
z
∩
y
=
∅
)
assuming
y
ↆ
u
∧¬
(
y
=
∅
)
∈{
1
,
1
/
2
}
.Let
ʳ
=min
{
ʲ
∈
ORD
|
there exists
t
∈
dom(
y
) such that
y
(
t
)
∈{
1
,
1
/
2
}
and
t
is
ʲ
-like
}
.
1. There exists
t
∈
dom(
y
) such that
y
(
t
)
By our assumption,
ʳ
≥
∈{
1
,
1
/
2
}
and
t
is
ʳ
-like.
∃
=
∃z
(
z ∈ y ∧¬∃w
(
w ∈ z ∧ w ∈ y
))
≥
t
∈ y ∧¬∃w
(
w ∈ t
∧ w ∈ y
))
=
t∈
dom(
y
)
z
(
z
∈
y
∧
z
∩
y
=
∅
)
(
y
(
t
)
∧
t
=
t
)
∧
(
w∈
dom(
t
)
(
t
(
w
)
∧
w ∈ y
))
∗
[
w∈
dom(
t
)
(
y
(
t
)
t
=
t
(
t
(
w
)
))]
∗
≥
∧
)
∧
∧
(
y
(
t
)
∧
w
=
t
t∈
dom(
y
)
[
w∈
dom(
t
)
(
t
(
w
)
))]
∗
.
≥
1
/
2
∧
∧
(
y
(
t
)
∧
w
=
t
t∈
dom(
y
)
5
Since PS
3
satisfies the deductive principle: ((
a ∧ b
)
⃒ c
)=(
a ⃒
(
b ⃒ c
)).
