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Generalized algebra-valued models of set theory
Löwe, B.; Tarafder, S.
DOI
10.1017/S175502031400046X
Publication date
2015
Document Version
Final published version
Published in
Review of Symbolic Logic
Link to publication
Citation for published version (APA):
Löwe, B., & Tarafder, S. (2015). Generalized algebra-valued models of set theory.
Review of
Symbolic Logic
,
8
(1), 192-205. https://doi.org/10.1017/S175502031400046X
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Download date:10 Aug 2022
T
HE
R
EVIEW OF
S
YMBOLIC
L
OGIC
Volume 8, Number 1, March 2015
GENERALIZED ALGEBRA-VALUED MODELS OF SET THEORY
BENEDIKT LÖWE
Institute for Logic, Language and Computation, Universiteit van Amsterdam and
Fachbereich Mathematik, Universität Hamburg
SOURAV TARAFDER
Department of Commerce (Morning), St. Xavier’s College and Department of Pure
Mathematics, Calcutta University
Abstract. We generalize the construction of lattice-valued models of set theory due to Takeuti,
Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a
paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel
set theory.
§1. Introduction. If B is any Boolean algebra and V a model of set theory, we can
construct by transfinite recursion the Boolean-valued model of set theory V
B
consisting of
names for sets, an extended language L
B
, and an interpretation function · : L
B
→ B
assigning truth values in B to formulas of the extended language. Using the notion of
validity derived from ·, all of the axioms of ZFC are valid in V
B
. Boolean-valued models
were introduced in the 1960s by Scott, Solovay, and Vop
ˇ
enka; an excellent exposition of
the theory can be found in Bell (2005).
Replacing the Boolean algebra in the above construction by a Heyting algebra H, one
obtains a Heyting-valued model of set theory V
H
. The proofs of the Boolean case transfer
to the Heyting-valued case to yield that V
H
is a model of IZF, intuitionistic ZF, where
the logic of the Heyting algebra H determines the logic of the Heyting-valued model of
set theory (cf. Grayson, 1979; Bell, 2005, chap. 8). This idea was further generalized
by Takeuti & Titani (1992), Titani (1999), Titani & Kozawa (2003), Ozawa (2007), and
Ozawa (2009), replacing the Heyting algebra H by appropriate lattices that allow models
of quantum set theory (where the algebra is an algebra of truth-values in quantum logic) or
fuzzy set theory.
In this paper, we shall generalize this model construction further to work on algebras that
we shall call reasonable implication algebras (§2). These algebras do not have a negation
symbol, and hence we shall be focusing on the negation-free fragment of first-order logic:
the closure under the propositional connectives ∧, ∨, ⊥, and →. Classically, of course,
every formula is equivalent to one in the negation-free fragment (since ¬ϕ is equivalent to
ϕ →⊥). In §3, we define the model construction and prove that assuming a number
of additional assumptions (among them a property we call the bounded quantification
property), we have constructed a model of the negation-free fragment of ZF
−
(which is
classically equivalent to ZF
−
).
In §4 and §5, we apply the results of §3 to a particular three-valued algebra where we
prove the bounded quantification property (§4) and the axiom scheme of Foundation (§5).
Received: June 20, 2014.
c
Association for Symbolic Logic, 2014
192
doi:10.1017/S175502031400046X
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GENERALIZED ALGEBRA
-
VALUED MODELS OF SET THEORY
193
Finally, in §6, we add a negation symbol to our language. With the appropriate negation,
our example from §4 and §5 becomes a model of a paraconsistent set theory that validates
all formulas from the negation-free fragment of ZF. We compare our paraconsistent set
theory to other paraconsistent set theories from the literature and observe that it is funda-
mentally different from them.
We should like to mention that Joel Hamkins independently investigated the construction
that is at the heart of this paper and proved a result equivalent to our Theorem 6.3 (presented
at the Workshop on Paraconsistent Set Theory in Storrs, CT in October 2013).
§2. Reasonable implication algebras.
Implication algebras and implication-negation algebras. In this paper, all structures
(A, ∧, ∨, 0, 1) will be complete distributive lattices with smallest element 0 and largest
element 1. As usual, we abbreviate x ∧ y = x as x ≤ y. An expansion of this structure by
an additional binary operation ⇒ is called an implication algebra and an expansion with
⇒ and another unary operation
∗
is called an implication-negation algebra. We emphasize
that no requirements are made for ⇒ and
∗
at this point.
Interpreting propositional logic in algebras. By L
Prop
we denote the language of
propositional logic without negation (with connectives ∧, ∨, →, and ⊥ and countably
many variables Var); we write L
Prop,¬
for the expansion of this language to include the
negation symbol ¬. Let L be either L
Prop
or L
Prop,¬
, and let A be either an implication
algebra or an implication-negation algebra, respectively. Any map ι from Var to A (called
an assignment) allows us to interpret L-formulas ϕ as elements ι(ϕ) of the algebra. Par
abus de langage, for an L-formula ϕ and some X ⊆ A, we write ϕ ∈ X for “for all
assignments ι :Var→ A, we have that ι(ϕ) ∈ X ”. As usual, we call a set D ⊆ A a filter if
the following four conditions hold: (i) 1 ∈ D, (ii) 0 /∈ D, (iii) if x, y ∈ D, then x ∧ y ∈ D,
and (iv) if x ∈ D and x ≤ y, then y ∈ D; in this context, we call filters designated sets of
truth values, since the algebra A and a filter D together determine a logic
A
,D
by defining
for every set of L
Prop
-formulas and every L
Prop
-formula ϕ
A
,D
ϕ : ⇐⇒ if for all ψ ∈ ,wehaveψ ∈ D, then ϕ ∈ D.
We write Pos
A
:={x ∈ A ; x = 0} for the set of positive elements in A. In all of the
examples considered in this paper, this set will be a filter.
The negation-free fragment. If L is any first-order language including the connectives
∧, ∨, ⊥ and → and any class of L-formulas, we denote closure of under ∧, ∨, ⊥,
∃, ∀, and → by Cl() and call it the negation-free closure of . A class of formulas
is negation-free closed if Cl() = . By NFF we denote the negation-free closure of the
atomic formulas; its elements are called the negation-free formulas.
1
Obviously, if L does not contain any connectives beyond ∧, ∨, ⊥, and →, then NFF =
L. Similarly, if the logic we are working in allows to define negation in terms of the other
connectives (as is the case, e.g., in classical logic), then every formula is equivalent to one
in NFF.
1
In some contexts, our negation-free fragment is called the positive fragment; in other contexts,
the positive closure is the closure under
∧, ∨, ⊥, ∃, and ∀ (not including →). In order to avoid
confusion with the latter contexts, we use the phrase “negation-free” rather than “positive”.
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S175502031400046X
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194
BENEDIKT LÖWE AND SOURAV TARAFDER
Reasonable implication algebras. We call an implication algebra A = (A, ∧, ∨, 0,
1, ⇒) reasonable if the operation ⇒ satisfies the following axioms:
P1 (x ∧ y) ≤ z implies x ≤ (y ⇒ z),
P2 y ≤ z implies (x ⇒ y) ≤ (x ⇒ z), and
P3 y ≤ z implies (z ⇒ x) ≤ (y ⇒ x).
We say that a reasonable implication algebra is deductive if
((x ∧ y) ⇒ z) = (x ⇒ (y ⇒ z)).
It is easy to see that any reasonable implication algebra satisfies that x ≤ y implies x ⇒
y =
1. Similarly, it is easy to see that in reasonable and deductive implication algebras, we
have (x ⇒ y) = (x ⇒ (x ∧ y)). These facts are being used in the calculations later in
the paper. It is easy to check that all Boolean algebras and Heyting algebras are reasonable
and deductive implication algebras.
Recurring examples. The following two examples will be crucial during the rest of
the paper: The three-valued Łukasiewicz algebra Ł
3
= ({0,
1
/
2
, 1}, ∧, ∨, ⇒, 0, 1) with
operations defined as in Figure 1 is a reasonable, but non-deductive implication algebra.
The three-valued algebra PS
3
= ({0,
1
/
2
, 1}, ∧, ∨, ⇒, 0, 1) with operations defined as in
Figure 2 is a reasonable and deductive implication algebra which is not a Heyting algebra.
Let us emphasize that, contrary to usage in other papers, we consider Ł
3
and PS
3
as
implication algebras without negation (cf. §6 for adding negations to PS
3
).
§3. The model construction.
3.1. Definitions and basic properties. Our construction follows very closely the
Boolean-valued construction as it can be found in Bell (2005). We fix a model of set theory
V and an implication algebra A = (A, ∧, ∨, 0, 1, ⇒) and construct a universe of names
by transfinite recursion:
V
A
α
={x ; x is a function and ran(x) ⊆ A
and there is ξ<αwith dom(x) ⊆ V
A
ξ
)} and
V
A
={x ;∃α(x ∈ V
A
α
)}.
We note that this definition does not depend on the algebraic operations in A, but only on
the set A, so any expansion of A to a richer language will give the same class of names
V
A
.ByL
∈
, we denote the first-order language of set theory using only the propositional
connectives ∧, ∨, ⊥, and →. We can now expand this language by adding all of the
Fig. 1. Connectives for the algebra Ł
3
.
Fig. 2. Connectives for PS
3
.
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GENERALIZED ALGEBRA
-
VALUED MODELS OF SET THEORY
195
elements of V
A
as constants; the expanded (class-sized) language will be called L
A
.Asin
the Boolean case (Bell, 2005, Induction Principle 1.7), the (meta-)induction principle for
V
A
can be proved by a simple induction on the rank function: for every property of
names, if for all x ∈ V
A
,wehave
∀y ∈ dom(x )((y)) implies (x),
then all names x ∈ V
A
have the property .
As in the Boolean case, we can now define a map · assigning to each negation-free
formula in L
A
a truth value in A as follows. If u,v in V
A
and ϕ,ψ ∈ NFF, we define
⊥ = 0,
u ∈ v =
x∈dom(v)
(v(x) ∧ x = u),
u = v =
x∈dom(u)
(u(x) ⇒ x ∈ v) ∧
y∈dom(v)
(v(y) ⇒ y ∈ u),
ϕ ∧ ψ = ϕ ∧ ψ,
ϕ ∨ ψ = ϕ ∨ ψ,
ϕ → ψ = ϕ ⇒ ψ,
∀xϕ(x) =
u∈V
A
ϕ(u), and
∃xϕ(x) =
u∈V
A
ϕ(u).
As usual, we abbreviate ∃x(x ∈ u ∧ ϕ(x)) by ∃x ∈ u ϕ(x) and ∀x(x ∈ u → ϕ(x)) by
∀x ∈ u ϕ(x) and call these bounded quantifiers. Bounded quantifiers will play a crucial
role in this paper.
If D is a filter on A and σ is a sentence of L
A
, we say that σ is D-valid in V
A
if σ ∈ D
and write V
A
|
D
σ .
In the Boolean-valued case, the names behave nicely with respect to their interpretations
as names for sets. For instance, if two names denote the same object, then the properties
of the object do not depend on the name you are using. In our generalized setting, we have
to be very careful since many of these reasonable rules do not hold in general: cf. §4 for
details.
P
ROPOSITION
3.1. If A is a reasonable implication algebra and u ∈ V
A
, we have that
u = u = 1 and u(x) ≤ x ∈ u (for each x ∈ dom(u)).
Proof. This is an easy induction, using the fact that we have that in all reasonable
implication algebras, x ≤ y implies x ⇒ y = 1.
However, things break down rather quickly if you go beyond Proposition 3.1. The in-
equality u = v ∧ v = w ≤ u = w representing transitivity of equality of names does
not hold in general in the model constructed over Ł
3
: consider the functions
p
0
={∅, 0},
p
1
/
2
={∅,
1
/
2
}, and
p
1
={∅, 1}.
Then it can be easily checked that p
0
= p
1
/
2
=
1
/
2
= p
1
/
2
= p
1
> p
0
= p
1
= 0.
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