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In order to show that an instance of bounded quantification over the formula
˕
behaves properly in
V
(
A
)
, one needs the validity of the formula BQ
˕
in
V
(
A
)
.
Unfortunately, the fact that
A
is a deductive reasonable implication algebra is
not sucient to prove this for all formulas
˕
; but it is sucient to prove it for
all negation-free formulas
˕
. Details can be found in [5,
§
3.1].
3
2.3 Definition of
α
-like Elements
As in the theory of Boolean valued models, we can define the notion of a
canonical
name
:
Definition 5.
For each x
∈
V
,
x
=
{
y,
1
:
y
∈
x
}
Let ORD refer to the class of all ordinal numbers in
V
. The main goal of
this paper is to identify elements in
V
(PS
3
)
which behave almost similar to the
classical ordinal numbers. It will be shown that there are more than one such
elements in
V
(PS
3
)
corresponding to each
ʱ
ORD which will be named as
ʱ
-
like elements. But the non-classical behaviour of these elements will be discussed
in
∈
4.
For each
ʱ
§
ORD the
ʱ
-like names in
V
(PS
3
)
are defined by transfinite
recursion as follows.
∈
V
(PS
3
)
is called
Definition 6.
An element x
∈
i)
0-like
if for every y
∈
dom(
x
)
, we have that x
(
y
)=0
;and
ii) ʱ
-like
if for each ʲ
∈
ʱ there exists y
∈
dom(
x
)
which is ʲ-like and x
(
y
)
∈
{
1
,
1
/
2
}
, and for any z
∈
dom(
x
)
if it is not ʲ-like for any ʲ
∈
ʱ then
x
(
z
)=0
.
Clearly, the canonical name
ʱ
is an
ʱ
-like name for every
ʱ
∈
ORD.
3 Properties of
α
-like Elements
For each
ʱ
∈
ORD, there are many
ʱ
-like names as the following results show.
V
(PS
3
)
and ʱ
Lemma 7.
For any x
∈
∈
ORD
,
x
=
ʱ
=1
if and only if x is
ʱ-like.
Proof.
The proof will be done by induction on the domain of
ʱ
. We assume that
we have shown the result for all elements in the domain of
ʱ
.Weknow
=
y∈
dom(
x
)
ʲ
x
=
ʱ
(
x
(
y
)
⃒
y
∈
ʱ
)
∧
(1
⃒
∈
x
)
ʲ
∈
dom(
ʱ
)
3
If
A
is a Boolean algebra or a Heyting algebra, then BQ
˕
canbeprovedforall
˕
.
