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In-Depth Information
Claim 12.
[
w∈
dom(
t
)
(
t
(
w
)
∧
t∈
dom(
y
)
(
y
(
t
)
∧
w
=
t
))]
∗
=1
.
Proof.
It is sucient to prove that
(
t
(
w
)
∧
(
y
(
t
)
∧
w
=
t
)) = 0
.
w∈
dom(
t
)
t∈
dom(
y
)
Assume for some
w
∈
dom(
t
),
t
(
w
)
∈{
1
,
1
/
2
}
. If possible let there exist
t
∈
dom(
y
) such that both
y
(
t
)
,
w
=
t
∈{
1
,
1
/
2
}
. By our assumption
y
(
t
)
∈{
1
,
1
/
2
}
implies
t
is
ʲ
-like for some
ʲ
∈
ʱ
.Since
w
=
t
∈{
1
,
1
/
2
}
by Theorem 9 we have
w
is
ʲ
-like. Again since
t
is
ʳ
like and
t
(
w
)
∈{
1
,
1
/
2
}
therefore
ʲ
∈
ʳ
which
contradicts the minimality of
ʳ
as
y
(
t
)
∈{
1
,
1
/
2
}
. Hence the claim is proved.
Therefore
∃
z
(
z
∈
y
∧
z
∩
y
=
∅
)
≥
1
/
2
∧
1=
1
/
2
. This leads to the fact that
V
(PS
3
)
,if
for any
y
∈
y
ↆ
u
∧¬
(
y
=
∅
)
∈{
1
,
1
/
2
}
then
∃
z
(
z
∈
y
∧
z
∩
y
=
∅
)
∈{
1
,
1
/
2
}
; i.e.,
∀
y
(
y
ↆ
u
∧¬
(
y
=
∅
)
→∃
z
(
z
∈
y
∧
z
∩
y
=
∅
))
=1
.
Hence we can conclude
V
(PS
3
)
Combining (
i
)and(
iii
) of lemma 11 the following theorem can be derived.
|
=WO
∈
(
u
).
Theorem 13.
Let ʱ ∈
ORD
and u be an ʱ-like element in
V
(PS
3
)
.Then
V
(PS
3
)
|
=ORD(
u
)
.
Theorem 13 shows any
ʱ
-like element satisfies the classical definition of ordinal
number. It is proved in [5] that the general Comprehension axiom scheme is not
valid in
V
(PS
3
)
. On the other hand it is a theorem of the paraconsistent set theory
considered in [12]. As a consequence the collection of all ordinals becomes a set in
that model. This fact leads us to the important question, whether the collection
of elements which make the first order formula ORD(
x
) valid is a set in
V
(PS
3
)
.
The following theorem assures the answer is negative.
Theorem 14.
There is no set of all ordinals:
V
(PS
3
)
∃
O
∀
x
(ORD(
x
)
→
x
∈
O
)
.
V
(PS
3
)
be arbitrarily chosen. Then by definition, dom(
O
)isa
set in
V
. By Theorem 9, if
ʱ
Proof.
Let
O
∈
=
ʲ
for any
ʱ, ʲ
∈
ORD then for any
ʱ
-like
u
and
ʲ
-like
v
,
V
(PS
3
)
u
=
v
. Hence
u
and
v
are not equal as a set in
V
.We
conclude that if for each
ʱ
ORD there exists an
ʱ
-like
u
in dom(
O
)then
dom(
O
) cannot be a set in
V
as the collection of all ordinals is not a set in
V
.
Hence there exists an
ʱ
∈
∈
ORD such that there is no
ʱ
-like element in dom(
O
).
Let
u
be an
ʱ
-like element. Then by Theorem 13,
ORD(
u
)
∈{
1
,
1
/
2
}
but
=
x∈
dom(
O
)
u
∈
O
(
O
(
x
)
∧
x
=
u
)=0
.
