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∀
x
(ORD(
x
)
→
x
∈
O
)
=0.Since
O
is arbitrary we have
Hence
∃
O
∀
x
(ORD(
x
)
→
x
∈
O
)
=0
.
So the theorem is proved.
Non-Classical Behavior of
α
-like Elements
Let us consider an
ʱ
ORD and a formula
˕
(
x
) with one free variable
x
.One
may expect for any two
ʱ
-like names
u
and
v
, they agree on whether the property
˕
holds of them.
But the ordinal-like elements behave non-classically: there exists a formula
˕
(
x
) such that for any given non-zero
ʱ
∈
∈
ORD there exist two
ʱ
-like elements
u
and
v
such that
V
(PS
3
)
=
˕
(
u
)whereas
V
(PS
3
)
|
˕
(
v
). For example, let
˕
(
x
):=
¬∃
y
(
y
∈
x
) and choose any non-zero
ʱ
∈
ORD. Fix any two
ʱ
-like elements
u
and
v
as
ran
(
u
)=
{
1
/
2
}
and
ran
(
v
)=
{
1
}
. Then clearly
˕
(
u
)
=
1
/
2
, i.e.,
V
(PS
3
)
= 0, i.e.,
V
(PS
3
)
˕
(
v
).
In [5], it is stressed that Leibniz's law of the indiscernibility of identicals
|
=
˕
(
u
). But it is easy to calculate that
˕
(
v
)
∀
x
∀
y
(
x
=
y
∧
˕
(
x
)
→
˕
(
y
))
is not necessarily valid in
V
(
A
)
if
A
is a reasonable implication algebra. The
above formula
˕
(
x
):=
x
) is one instance of that claim.
On the other hand it is also proved that for any instantiations of Leibniz's
law with NFF-formulas
˕
is valid in general, and so
ʱ
-like names agree on the
validity of NFF-formulas.
¬∃
y
(
y
∈
5 Conclusion and Future Work
We have seen that the
ʱ
-like names form equivalence classes in
V
(PS
3
)
/
.How-
ever, due to the failure of Leibniz's law in
V
(PS
3
)
, elements in the same
∼
∼
-
equivalence class can instantiate different properties.
In future work, we plan to study the
natural numbers
,
rational numbers
,and
real numbers
in
V
(
A
)
, together with their algebraic properties, as well as cardinal
numbers in
V
(
A
)
.
Acknowledgements.
The author would like to thank Benedikt Lowe for the
collaboration during his stay at the
Universiteit van Amsterdam
in June 2014
that was the basis of several ideas in this paper. The visit to Amsterdam was
made possible by a travel grant from the
Indo-European Research Training Net-
work in Logic (IERTNiL)
. He is also thankful to Mihir Kr. Chakraborty for
providing important suggestions in some deeper issues and in shaping the final
version of this paper. The author's research is partially funded by
CSIR
,Ph.D.
fellowship, Govt. of India (09/028(0754)/2009-EMR-I).
