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but along with these tables there is one more table for an unary operator called
the
inconsistency operator
. Also the approaches there are totally different from
[9] which was developed independently.
In this paper, we study the set theory of the model
V
(PS
3
)
, in particular
the notion of ordinals in this particular paraconsistent set theory. Ordinals have
been studied in non-classical set theories; e.g., Titani used a notion of ordinal
number in
lattice-valued set theory
[10] where the definition of ordinal is not
exactly the one used in classical set theory. On the other hand let us consider
the paraconsistenet set theory described by [12] where the classical definition of
ordinal is used.
In
4, we stick to the classical definition of ordinals. However, our paracon-
sistent set theory differs considerably from Weber's (as noted in [5]) since the
axiom scheme of full comprehension is valid in Weber's set theory and invalid in
V
(PS
3
)
.
2
Usually in the literature of paraconsistent set theory general comprehension
is taken as valid. In [5] it is proved that
V
(PS
3
)
does not validate the general
comprehension axiom scheme. Also theorem 14 of this paper shows that there
does not exist a set of all ordinal numbers which disproves the general com-
prehension. One of the motivations of this paper in connection with the work
done in [5] is that without the comprehension axiom scheme a set theory may
behave well enough as a paraconsistent set theory. There is no need to think
that paraconsistent set theories are only built to deal with the set theoretic
paradoxes e.g.,
Russell's paradox
. Rather this paper will show a set theory being
paraconsistent may agree with important classical set theoretic results. This fact
is corroborated in Intuitionistic set theory as viewed in Heyting algebra-valued
models [4]. It may also be mentioned that the identity as used here is not exactly
the classical one. In future we shall deal with these issues in more detail.
§
2 Definitions and Preliminaries
2.1 Some Classical Definitions
We develop the classical theory of
transitive sets
,
well-ordered sets
, and ordi-
nal numbers in the setting of
V
(
A
)
. The following definitions 2-4 are stated in
classical metalanguage whereas the formalizations will take place in
§
4.
Definition 2.
Asetx is said to be transitive if every element of x is a subset
of x, or equivalently, if y
∈
z and z
∈
x implies y
∈
x.
Definition 3.
Aset
A
is said to be well-ordered by a relation
R
if
R
is a linear
order on
A
and any non-empty subset of
A
has a least element with respect to
R
.
Definition 4.
An ordinal number is a transitive set well-ordered by
∈
.
2
Comprehension axiom scheme: if P(
x
) is a property then
{x
:P(
x
)holds
}
is a set.
