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Ordinals in an Algebra-Valued Model
of a Paraconsistent Set Theory
Sourav Tarafder
1
,
2
1
Department of Commerce,
St. Xavier's College,
30, Mother Teresa Sarani, Kolkata, 700016, India
2
Department of Pure Mathematics,
University of Calcutta,
35, Ballygunge Circular Road, Kolkata, 700019, India
souravt09@gmail.com
Abstract.
This paper deals with ordinal numbers in an algebra-valued
model of a paraconsistent set theory. It is proved that the collection of
all ordinals is not a set in this model which is dissimilar to the other
existing paraconsistent set theories. For each ordinal
ʱ
of classical set
theory
ʱ
-like elements are defined in the mentioned algebra-valued model
whose collection is not singleton. It is shown that two
ʱ
-like elements
(for same
ʱ
) may perform conversely to validate a given formula of the
corresponding paraconsistent set theory.
Keywords:
non-classical set theory, ordinal numbers, paraconsistent
logic.
1
Introduction
Boolean-valued models
of classical set theory were introduced by Dana Scott,
Robert M. Solovay and Petr Vopenka in the 1960s. If
B
is a
complete Boolean
algebra
then
V
(
B
)
is a model of Zermelo Fraenkel set theory with the Axiom of
Choice, ZFC (cf. [1]). If the complete Boolean algebra is replaced by a
complete
Heyting algebra
,
then essentially the same proof shows that
V
(
H
)
becomes
a model of Intuitionistic Zermelo Fraenkel set theory, IZF [4]. Later Takeuti,
Titani, Kozawa and Ozawa generalised the development to some appropriate
lattice-valued model
of
quantum set theory
or
fuzzy set theory
[6-8] [10, 11].
We describe the construction of these algebra-valued models and the notion of
validity of a formula in these models in
H
2.
It can be proved that for Heyting-valued models
V
(
H
)
, the validity of the
Axiom of Choice AC in
V
(
H
)
is equivalent to the
law of excluded middle a
§
∨
a
∗
in
. This is a remarkable fact linking set theoretic properties in algebra-
valued models of set theories to algebraic properties of the corresponding algebra.
This observation was used in [5] to define a class of algebras
H
A
,
∧
,
∨
,
⃒
,
1
,
0
called reasonable implication algebras. An algebra
A
,
∧
,
∨
,
⃒
,
1
,
0
is called a
reasonable implication algebra
if the following hold:
