Information Technology Reference
In-Depth Information
where the following abbreviations are used in WO
∈
(
x
):
y
ↆ
x
:=
∀
t
(
t
∈
y
→
t
∈
x
)
,
¬
(
y
=
∅
):=
∃
z
(
z
∈
y
)
,
(
z
∩
y
=
∅
):=
¬∃
w
(
w
∈
z
∧
w
∈
y
)
.
Finally, we can connect the notion of
ʱ
-like name to the set theoretic notion
of ordinals:
Lemma 11.
Let ʱ
∈
ORD
and u be an ʱ-like element in
V
(PS
3
)
. Then the
following hold:
i)
V
(PS
3
)
|
=Trans(
u
)
ii)
V
(PS
3
)
|
=LO(
u
)
iii)
V
(PS
3
)
|
=WO
∈
(
u
)
Proof.
(
i
)Wehavetoprove
∀
y
∀
z
(
z
∈
y
∧
y
∈
u
→
z
∈
u
)
∈{
1
,
1
/
2
}
. Since the
truth table of
⃒
in PS
3
does not contain
1
/
2
it is sucient to show
∀
y
∀
z
(
z
∈
y
∧
y
∈
u
→
z
∈
u
)
=1.
V
(PS
3
)
. Then,
Let us take any
z
∈
=
y∈
V
(PS
3
)
∀
y
(
y
∈
u
∧
z
∈
y
→
z
∈
u
)
(
y
∈
u
∧
z
∈
y
⃒
z
∈
u
)
=
y∈
V
(PS
3
)
y
∈
u
⃒
z
∈
y
⃒
z
∈
u
(
(
))
=
y∈
dom(
u
)
(
u
(
y
)
⃒
z
∈
y
⃒
z
∈
u
(
))
(since BQ
˕
hold in
V
(PS
3
)
for all negation-free formulas
˕
.)
For any
y
∈
dom(
u
)if
u
(
y
)
=0then
y
is
ʲ
-like for some non-zero
ʲ
∈
ʱ
.Let
for such an
y
,
z
∈
y
∈{
1
,
1
/
2
}
. Therefore by Theorem 10,
z
is
ʳ
-like for some
ʳ
∈
ʲ
. Clearly,
ʳ
∈
ʱ
. Therefore one more application of Theorem 10 provides
z
∈
u
∈{
1
,
1
/
2
}
. Hence combining the above results we get
(
u
(
y
)
⃒
(
z
∈
y
⃒
z
∈
u
)) = 1
y∈
dom(
u
)
∈
V
(PS
3
)
. This leads to the fact
for any
z
=1
,
i.e.,
V
(PS
3
)
∀
y
∀
z
(
y
∈
u
∧
z
∈
y
→
z
∈
u
)
|
=Trans(
u
)
.
(
ii
)Sinceforany
ʱ, ʲ
∈
ORD exactly one of
ʱ
∈
ʲ, ʱ
=
ʲ
and
ʲ
∈
ʱ
holds
in
V
, the proof can be derived easily by applying Theorems 9 and 10.
