2.1 Generalized algebra-valued models

Let $\Lambda $ be a set of logical connectives; we shall assume that

$$ \begin{align*}\{\wedge,\vee,\top, \bot\}\subseteq \Lambda\subseteq\{\wedge,\vee,\to,\neg,\top, \bot\},\end{align*} $$

where $\wedge ,\vee $ , and $\to $ are binary connectives; $\neg $ is a unary connective; $\top $ and $\bot $ are two $0$ -ary connectives.

Fix a countable set of propositional variables, that we call $\mathrm {Prop}$ , and a countable set of first-order variables, that we call $\mathrm {Var}$ . The propositional logic with propositional variables in $\mathrm {Prop}$ and connectives in $\Lambda $ will be denoted by $\mathcal {L}_\Lambda $ . The first-order logic of set theory with variables in $\mathrm {Var}$ , the binary predicate symbol $\in $ , and propositional connectives from $\Lambda $ will be denoted by $\mathcal {L}_{\Lambda ,\in }$ . The set of sentences of $\mathcal {L}_{\Lambda ,\in }$ will be denoted by $\mathrm {Sent}_{\Lambda ,\in }$ . Observe that both $\mathcal {L}_\Lambda $ and $\mathrm {Sent}_{\Lambda ,\in }$ have the same structure of $\Lambda $ -algebra, for a fixed $\Lambda $ . This fact will be used in Section §4.1 to define homomorphisms between these structures.

For a set of logical connectives $\Lambda $ , we define $\mathrm {NFF}_{\Lambda , \in }$ to be the closure of the atomic formulas in $\mathcal {L}_{\Lambda , \in }$ under the connectives in $\Lambda $ other than the connective $\neg $ . It might be the case that $\Lambda $ does not contain $\neg $ , in which case $\mathrm {NFF}_{\Lambda , \in }$ will be same as $\mathcal {L}_{\Lambda , \in }$ . Since, any formula $\neg \varphi $ is classically (intuitionistically) equivalent to $\varphi \to \bot $ , $\mathrm {NFF}_{\Lambda , \in }$ and $\mathcal {L}_{\Lambda , \in }$ are equivalent in strength in first-order classical (intuitionistic) logic. If the set of connectives is clear from the context, we shall denote $\mathrm {NFF}_{\Lambda , \in }$ by $\mathrm {NFF}$ only. The formulas in $\mathrm {NFF}$ are called the negation-free formulas. By $\mathrm {NFF}$ - $\mathsf {ZF}$ and $\mathrm {NFF}$ - $\mathsf {ZF}^-$ we mean the negation free fragment of $\mathsf {ZF}$ and the negation free fragment of $\mathsf {ZF}$ excluding the $\mathsf {Foundation\ Axiom}$ ,Footnote ⁵ respectively.

Consider a model $\mathbf {V}$ of $\mathsf {ZFC}$ and a complete $\Lambda $ -algebra $\mathbb {A} = \langle \mathbf {A}, \wedge , \vee , \Rightarrow , ^*, \mathbf {1}, \mathbf {0}\rangle $ , where

1. (i) $\Lambda = \{\wedge ,\vee ,\to ,\neg ,\top , \bot \}$ ,

2. (ii) the operators $\wedge $ , $\vee $ , $\Rightarrow $ , and $^*$ of $\mathbb {A}$ correspond to the connectives $\wedge $ , $\vee $ , $\to $ , and $\neg $ of $\Lambda $ , respectively,

3. (iii) the constants $\mathbf {1}$ and $\mathbf {0}$ of $\mathbb {A}$ correspond to the $0$ -ary connectives $\top $ and $\bot $ of $\Lambda $ , respectively.

A universe of $\mathbb {A}$ -names is constructed by transfinite recursion:

$$ \begin{align*} \mathbf{V}_{\alpha}^{(\mathbb{A})} &= \{x\,:\, x\ \mbox{is a function and }\mathrm{ran}(x) \subseteq \mathbf{A}\\[-3pt] &\hspace{15mm}\mbox{ and there is}\ \xi < \alpha\ \mbox{with }\mathrm{dom}(x) \subseteq \mathbf{V}_{\xi}^{(\mathbb{A})})\},\mbox{ and}\\[-3pt] \mathbf{V}^{(\mathbb{A})} &= \{x\,:\,\exists \alpha (x \in \mathbf{V}_{\alpha}^{(\mathbb{A})})\}. \end{align*} $$

Let $\mathcal {L}_{\mathbb {A}}$ stand for the logic in the extended language of $\mathcal {L}_{\Lambda , \in }$ , extended by adding constants corresponding to each element in $\mathbf {V}^{(\mathbb {A})}$ .

Following the Boolean-valued model construction a map $[\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}}$ is defined from the class of all formulas in the extended language to the set $\mathbf {A}$ of truth values as follows. If $u,v \in \mathbf {V}^{(\mathbb {A})}$ and $\varphi , \psi $ are any two formulas, then

$$ \begin{align*} [\kern-1.5pt[ \top ]\kern-1.5pt]_{\mathbb{A}} & = \mathbf{1},\\[-0.5pt] [\kern-1.5pt[ \bot ]\kern-1.5pt]_{\mathbb{A}} & = \mathbf{0},\\[-0.5pt] [\kern-1.5pt[ u \in v ]\kern-1.5pt]_{\mathbb{A}} &= \bigvee_{x \in \mathrm{dom}(v)} (v(x) \land [\kern-1.5pt[ x = u ]\kern-1.5pt]_{\mathbb{A}}),\\[-0.5pt] [\kern-1.5pt[ u = v ]\kern-1.5pt]_{\mathbb{A}} &= \bigwedge_{x \in \mathrm{dom}(u)} (u(x) \Rightarrow [\kern-1.5pt[ x \in v ]\kern-1.5pt]_{\mathbb{A}}) \land \bigwedge_{y \in \mathrm{dom}(v)} (v(y) \Rightarrow [\kern-1.5pt[ y \in u ]\kern-1.5pt]_{\mathbb{A}} ),\\ [\kern-1.5pt[ \varphi \land \psi ]\kern-1.5pt]_{\mathbb{A}} & = [\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{A}} \land [\kern-1.5pt[ \psi ]\kern-1.5pt]_{\mathbb{A}},\\ [\kern-1.5pt[ \varphi \lor \psi ]\kern-1.5pt]_{\mathbb{A}} & = [\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{A}} \lor [\kern-1.5pt[ \psi ]\kern-1.5pt]_{\mathbb{A}},\\ [\kern-1.5pt[ \varphi \rightarrow \psi ]\kern-1.5pt]_{\mathbb{A}} & = [\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{A}} \Rightarrow [\kern-1.5pt[ \psi ]\kern-1.5pt]_{\mathbb{A}},\\ [\kern-1.5pt[ \neg \varphi ]\kern-1.5pt]_{\mathbb{A}} & = [\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{A}}^*,\\[-0.5pt] [\kern-1.5pt[ \forall x \varphi(x) ]\kern-1.5pt]_{\mathbb{A}} &= \bigwedge_{u \in \mathbf{V}^{(\mathbb{A})}} [\kern-1.5pt[ \varphi(u) ]\kern-1.5pt]_{\mathbb{A}}\mbox{, and}\\[-0.5pt] [\kern-1.5pt[ \exists x \varphi(x) ]\kern-1.5pt]_{\mathbb{A}} &= \bigvee_{u \in \mathbf{V}^{(\mathbb{A})}} [\kern-1.5pt[ \varphi(u) ]\kern-1.5pt]_{\mathbb{A}}. \end{align*} $$

Let $D\subseteq \mathbf {A}$ be a designated set. A formula $\varphi $ of $\mathcal {L}_{\mathbb {A}}$ is said to be D-valid in $\mathbf {V}^{(\mathbb {A})}$ if $[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {A}} \in D$ and is denoted by $\mathbf {V}^{(\mathbb {A})} \models _D \varphi $ . Abusing the notations, sometimes we shall denote the map $[\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}}$ by $[\kern-1.5pt[ \cdot ]\kern-1.5pt] $ and the validity relation $\mathbf {V}^{(\mathbb {A})} \models _D \varphi $ by $\mathbf {V}^{(\mathbb {A})} \models \varphi $ when the algebra $\mathbb {A}$ and the designated set D are clear from the context.

It is well-known that if $\mathbb {A}$ is a Boolean algebra or Heyting algebra then $\mathbf {V}^{(\mathbb {A})} \models \mathrm {ZF}$ , also in particular if $\mathbb {A}$ is a Boolean algebra then we get $\mathbf {V}^{(\mathbb {A})} \models \mathsf {Axiom \; of \; Choice}$ (cf. [Reference Bell3, Reference Grayson, Fourman, Mulvey and Scott6]).

Bounded quantification in the algebra-valued models

Let us consider a $\Lambda $ -algebra $\mathbb {A}$ , a formula $\varphi (x)$ in $\mathcal {L}_{\mathbb {A}}$ and an element $u \in \mathbf {V}^{(\mathbb {A})}$ . Then, the formula $\forall x (x \in u \to \varphi (x))$ is a bounded quantification over the formula $\varphi (x)$ . Following the definition of the map $[\kern-1.5pt[ \cdot ]\kern-1.5pt] $ , we have

$$\begin{align*}[\kern-1.5pt[\forall x(x\in u \to \varphi(x))]\kern-1.5pt] = \bigwedge_{x\in \mathbf{V}^{(\mathbb{A})}} (u(x)\Rightarrow [\kern-1.5pt[\varphi(x)]\kern-1.5pt]).\end{align*}$$

For any formula $\varphi (x)$ in $\mathcal {L}_{\mathbb {A}}$ , consider the following equation:

(BQ_φ) $$\begin{align} [\kern-1.5pt[\forall x(x\in u \to \varphi(x))]\kern-1.5pt] = \bigwedge_{x\in\mathrm{dom}(u)} (u(x)\Rightarrow [\kern-1.5pt[\varphi(x)]\kern-1.5pt]). \end{align}$$

If $\mathbb {A}$ is a Boolean algebra (or Heyting algebra) then it can be proved that for any formula $\varphi (x)$ and any $u \in \mathbf {V}^{(\mathbb {A})}$ ,

$$\begin{align*}\bigwedge_{x\in \mathbf{V}^{(\mathbb{A})}} (u(x)\Rightarrow [\kern-1.5pt[\varphi(x)]\kern-1.5pt]) = \bigwedge_{x\in\mathrm{dom}(u)} (u(x)\Rightarrow [\kern-1.5pt[\varphi(x)]\kern-1.5pt]),\end{align*}$$

which implies that $\mathrm {BQ}_\varphi $ holds in $\mathbf {V}^{(\mathbb {A})}$ for every formula $\varphi $ [Reference Bell3, corollary 1.18]. But there exists $\Lambda $ -algebra $\mathbb {A}$ and formulas $\varphi $ in $\mathcal {L}_{\mathbb {A}}$ such that $\mathrm {BQ}_\varphi $ does not hold in $\mathbf {V}^{(\mathbb {A})}$ [Reference Löwe and Tarafder13, p. 196].

For a given $\Lambda $ - algebra $\mathbb {A}$ , we say that the bounded quantification property holds for a formula $\varphi $ of $\mathcal {L}_{\mathbb {A}}$ if $\mathrm {BQ}_\varphi $ holds in $\mathbf {V}^{(\mathbb {A})}$ . It will said to be that the $\mathrm {NFF}$ -bounded quantification property ( $\mathrm {NFF}$ - $\mathrm {BQ}_\varphi $ ) holds in $\mathbf {V}^{(\mathbb {A})}$ if the bounded quantification property $\mathrm {BQ}_\varphi $ hold in $\mathbf {V}^{(\mathbb {A})}$ for all negation free formulas $\varphi $ . We will heavily depend on this property to establish the results throughout this paper.

2.2 Reasonable implication algebra $(\mathrm {RIA})$

The notion of reasonable implication algebra was first introduced in [Reference Löwe and Tarafder13] to develop a theory on generalized algebra-valued models which validate a ‘reasonable’Footnote ⁶ fragment of $\mathsf {ZF}$ , viz. $\mathrm {NFF}$ - $\mathsf {ZF}^-$ .

Definition 2.4 [Reference Löwe and Tarafder13, p. 194].

A complete distributive lattice, augmented with an operation $\Rightarrow $ , $\mathbb {A} := \langle \mathbf {A}, \wedge , \vee , \Rightarrow , \mathbf {1}, \mathbf {0} \rangle $ is called a reasonable implication algebra $(\mathrm {RIA})$ if the following properties hold:

1. P1. $(x\wedge y)\leq z$ implies $x\leq (y\Rightarrow z)$ ,

2. P2. $y\leq z$ implies $(x\Rightarrow y)\leq (x\Rightarrow z)$ , and

3. P3. $y\leq z$ implies $(z\Rightarrow x)\leq (y\Rightarrow x)$ .

A reasonable implication algebra is said to be deductive if, in addition,

(P4) $$\begin{align} ((x\wedge y)\Rightarrow z)=(x\Rightarrow (y\Rightarrow z)). \end{align}$$

2.3 A deductive $\mathrm {RIA}$ , $\mathrm {PS}_3$

As an example of a deductive reasonable implication algebra, beside Heyting and Boolean, we find a three-valued algebra with operations defined in Table 1 and supplemented with a unary operator $^*$ defined by $\mathbf {1}^* = \mathbf {0}$ , , and $\mathbf {0}^* = \mathbf {1}$ . We use the symbol $\mathrm {PS}_3$ to refer to the augmented structure $\langle \mathbb {PS}_3, ^* \rangle $ . The designated set is taken to be . In [Reference Tarafder, Chakraborty, Beziau, Chakraborty and Dutta18] a propositional logic $\mathbb {L}\mathrm {PS}_3$ is developed which is sound and (weak) complete with respect to $\mathrm {PS}_3$ . The axioms of $\mathbb {L}\mathrm {PS}_3$ are theorems of the classical propositional logic as well. Theorem 2.6 explains precisely the connection between $\mathbb {L}\mathrm {PS}_3$ and the classical propositional logic.

Table 1 Truth tables of the operations of $\mathbb {PS}_3$

It was proved in [Reference Löwe and Tarafder13] that for any negation free formula $\varphi $ , $\mathbf {V}^{(\mathrm {PS}_3)}$ satisfies $\mathrm {BQ}_\varphi $ . Moreover the negation free fragment of $\mathsf {Foundation \; Axiom}$ is valid in $\mathbf {V}^{(\mathrm {PS}_3)}$ . Hence, combining these results and using Theorem 2.5 we have the following theorem.

As a consequence $\mathbf {V}^{(\mathrm {PS}_3)}$ becomes an algebra-valued model for a paraconsistent set theory, which, however, differs from the classical $\mathsf {ZF}$ , since some instances of the axiom schemata of $\mathsf {ZF}$ fail in $\mathbf {V}^{(\mathrm {PS}_3)}$ .

Proof. Consider the following two $\mathrm {PS}_3$ -names: $u=\{\langle \varnothing , \mathbf {1} \rangle \}$ and

. Then,

by the definitions of implication and equality.

Consider the formula $\varphi (x) := \neg \exists y (y \in x)$ . Now,

Similarly, since $u(\varnothing ) = \mathbf {1} = [\kern-1.5pt[ \varnothing = \varnothing ]\kern-1.5pt] _{\mathrm {PS}_3}$ , we calculate $[\kern-1.5pt[ \varphi (u) ]\kern-1.5pt] _{\mathrm {PS}_3} = \mathbf {0}$ , which implies that $[\kern-1.5pt[ \varphi (u) ]\kern-1.5pt] _{\mathrm {PS}_3} \notin D_{\mathrm {PS}_3}$ . We go on to show that $\mathsf {Separation}$ fails. Consider the $\mathrm {PS}_3$ -names $u,v$ and the formula $\varphi (x)$ , as defined above and fix an element w of $\mathbf {V}^{(\mathrm {PS}_3)}$ as $w= \{\langle u, \mathbf {1} \rangle , \langle v, \mathbf {1} \rangle \}$ . In particular, we show that:

$$ \begin{align*} \bigvee_{y \in \mathbf{V}^{\mathrm{PS}_3}} \big{(} &\bigwedge_{x \in \mathbf{V}^{\mathrm{PS}_3}} ([\kern-1.5pt[ x \in y ]\kern-1.5pt]_{\mathrm{PS}_3} \Rightarrow ([\kern-1.5pt[ \varphi(x) ]\kern-1.5pt]_{\mathrm{PS}_3} \wedge [\kern-1.5pt[ x \in w ]\kern-1.5pt]_{\mathrm{PS}_3} )) \wedge \\ &\bigwedge_{x \in \mathbf{V}^{\mathrm{PS}_3}} ( ([\kern-1.5pt[ x \in w ]\kern-1.5pt]_{\mathrm{PS}_3} \wedge [\kern-1.5pt[ \varphi(x) ]\kern-1.5pt]_{\mathrm{PS}_3}) \Rightarrow [\kern-1.5pt[ x \in y]\kern-1.5pt]_{\mathrm{PS}_3})\big{)} \\ & \notin D_{\mathrm{PS}_3}. \end{align*} $$

Now suppose that, for an arbitrary $y_0 \in \mathbf {V}^{\mathrm {PS}_3}$ we have

$$ \begin{align*}\bigwedge_{x \in \mathbf{V}^{\mathrm{PS}_3}}( [\kern-1.5pt[ x \in w ]\kern-1.5pt]_{\mathrm{PS}_3} \wedge [\kern-1.5pt[ \varphi(x) ]\kern-1.5pt]_{\mathrm{PS}_3} \Rightarrow [\kern-1.5pt[ x \in y_0 ]\kern-1.5pt]_{\mathrm{PS}_3}) \in D_{\mathrm{PS}_3}.\end{align*} $$

In particular $\big {(}[\kern-1.5pt[ v \in w ]\kern-1.5pt] _{\mathrm {PS}_3} \wedge [\kern-1.5pt[ \varphi (v) ]\kern-1.5pt] _{\mathrm {PS}_3} \Rightarrow [\kern-1.5pt[ v \in y_0 ]\kern-1.5pt] _{\mathrm {PS}_3} \big {)} \in D_{\mathrm {PS}_3}$ . Since $[\kern-1.5pt[ v \in w ]\kern-1.5pt] _{\mathrm {PS}_3} = \mathbf {1}$ and $[\kern-1.5pt[ \varphi (v) ]\kern-1.5pt] _{\mathrm {PS}_3} \in D_{\mathrm {PS}_3}$ , we have $ [\kern-1.5pt[ v \in y_0 ]\kern-1.5pt] _{\mathrm {PS}_3} \in D_{\mathrm {PS}_3} $ . Therefore, there exists a $z_0 \in \mathrm {dom}(y_0)$ such that $ y_0(z_0) \wedge [\kern-1.5pt[ v = z_0 ]\kern-1.5pt] _{\mathrm {PS}_3} \in D_{\mathrm {PS}_3}$ . So we get $[\kern-1.5pt[ u = v ]\kern-1.5pt] _{\mathrm {PS}_3} \wedge [\kern-1.5pt[ v = z_0 ]\kern-1.5pt] _{\mathrm {PS}_3} \in D_{\mathrm {PS}_3}$ and thus $[\kern-1.5pt[ u = z_0 ]\kern-1.5pt] _{\mathrm {PS}_3} \in D_{\mathrm {PS}_3}$ . This implies $[\kern-1.5pt[ u \in y_0 ]\kern-1.5pt] _{\mathrm {PS}_3} \in D_{\mathrm {PS}_3}$ . But then since $[\kern-1.5pt[ \varphi (u) ]\kern-1.5pt] _{\mathrm {PS}_3} = \mathbf {0}$ we have:

$$\begin{align*}[\kern-1.5pt[ u \in y_0 ]\kern-1.5pt]_{\mathrm{PS}_3} \Rightarrow ([\kern-1.5pt[ \varphi(u) ]\kern-1.5pt]_{\mathrm{PS}_3} \wedge [\kern-1.5pt[ u \in w ]\kern-1.5pt]_{\mathrm{PS}_3} ) = \mathbf{0}. \end{align*}$$

Thus, for any $y \in \mathbf {V}^{(\mathrm {PS}_3)}$ if

$$\begin{align*}\bigwedge_{x \in \mathbf{V}^{\mathrm{PS}_3}} ( ([\kern-1.5pt[ x \in w ]\kern-1.5pt]_{\mathrm{PS}_3} \wedge [\kern-1.5pt[ \varphi(x) ]\kern-1.5pt]_{\mathrm{PS}_3}) \Rightarrow [\kern-1.5pt[ x \in y]\kern-1.5pt]_{\mathrm{PS}_3}) \neq \mathbf{0},\end{align*}$$

then,

$$ \begin{align*} \bigwedge_{x \in \mathbf{V}^{\mathrm{PS}_3}} [\kern-1.5pt[ x \in y ]\kern-1.5pt]_{\mathrm{PS}_3} \Rightarrow ([\kern-1.5pt[ \varphi(x) ]\kern-1.5pt]_{\mathrm{PS}_3} \wedge [\kern-1.5pt[ x \in w ]\kern-1.5pt]_{\mathrm{PS}_3}) = \mathbf{0}, \end{align*} $$

i.e.,

$$ \begin{align*} \bigvee_{y \in \mathbf{V}^{\mathrm{PS}_3}} \big{(} &\bigwedge_{x \in \mathbf{V}^{\mathrm{PS}_3}} ([\kern-1.5pt[ x \in y ]\kern-1.5pt]_{\mathrm{PS}_3} \Rightarrow ([\kern-1.5pt[ \varphi(x) ]\kern-1.5pt]_{\mathrm{PS}_3} \wedge [\kern-1.5pt[ x \in w ]\kern-1.5pt]_{\mathrm{PS}_3} )) \wedge \\ &\bigwedge_{x \in \mathbf{V}^{\mathrm{PS}_3}} ( ([\kern-1.5pt[ x \in w ]\kern-1.5pt]_{\mathrm{PS}_3} \wedge [\kern-1.5pt[ \varphi(x) ]\kern-1.5pt]_{\mathrm{PS}_3}) \Rightarrow [\kern-1.5pt[ x \in y]\kern-1.5pt]_{\mathrm{PS}_3})\big{)} \\ & = \mathbf{0}. \end{align*} $$

And this concludes the proof.□

For latter use, let us give the name $\mathsf {Sep}$ to the instance of the $\mathsf {Separation\ Axiom}$ that Theorem 2.8 shows to fail in $\mathbf {V}^{(\mathrm {PS}_3)}$ :

(Sep) $$ \begin{align} \forall x \exists y \forall z \big(z \in y \leftrightarrow (z \in x \land (\neg \exists w (w \in z)))\big).\nonumber \end{align} $$

3.1 Product of two algebras

In order to extend the class of algebras which give rise to algebra-valued models of set theories we shall combine them, using products. From now on, unless otherwise stated, we fix the following signature $\Lambda = \{\wedge , \vee , \to , \lnot , \top , \bot \}$ .

Definition 3.9. Let us consider two $\Lambda $ -algebras $\mathbb {A} = \langle \mathbf {A}, \wedge _{\mathbf {A}}, \vee _{\mathbf {A}}, \Rightarrow _{\mathbf {A}}, ^{*_{\mathbf {A}}}, \mathbf {1}_{\mathbf {A}}, \mathbf {0}_{\mathbf {A}}\rangle $ and $\mathbb {B} = \langle \mathbf {B}, \wedge _{\mathbf {B}}, \vee _{\mathbf {B}}, \Rightarrow _{\mathbf {B}}, ^{*_{\mathbf {B}}}, \mathbf {1}_{\mathbf {B}}, \mathbf {0}_{\mathbf {B}}\rangle $ . The product algebra $\mathbb {A} \times \mathbb {B}$ is the structure $\langle \mathbf {A} \times \mathbf {B}, \wedge\!, \vee , \Rightarrow , ^*, \mathbf {1}, \mathbf {0} \rangle $ having domain $\mathbf {A} \times \mathbf {B}$ and with operations defined coordinate-wise: i.e., for any $a, c \in \mathbf {A}$ and $b, d \in \mathbf {B},$

$$ \begin{align*} (a, b) \wedge (c, d) & = (a \wedge_{\mathbf{A}} c, b \wedge_{\mathbf{B}} d),\\(a, b) \vee (c, d) & = (a \vee_{\mathbf{A}} c, b \vee_{\mathbf{B}} d),\\(a, b) \Rightarrow (c, d) & = (a \Rightarrow_{\mathbf{A}} c, b \Rightarrow_{\mathbf{B}} d),\\(a, b)^* & = (a^{*_{\mathbf{A}}}, b^{*_{\mathbf{B}}}),\\\mathbf{1} & = (\mathbf{1}_{\mathbf{A}}, \mathbf{1}_{\mathbf{B}}), \mbox{ and}\\\mathbf{0} & = (\mathbf{0}_{\mathbf{A}}, \mathbf{0}_{\mathbf{B}}). \end{align*} $$

Observation 3.10. From the definition it follows that $\mathbb {A} \times \mathbb {B}$ is also a $\Lambda $ -algebra having the following property: for any two elements $(a,b), (c,d) \in \mathbb {A} \times \mathbb {B}$ ,

$$\begin{align*}(a, b) \leq (c, d) \mbox{ iff } a \leq_{\mathbf{A}} c \mbox{ and } b \leq_{\mathbf{B}} d.\end{align*}$$

Notice that, for each element $(a, b) \in \mathbb {A} \times \mathbb {B}$ , we have $\mathbf {0} \leq (a,b) \leq \mathbf {1}$ .

3.2 Algebra-valued models using the product algebras

In this paper, unless otherwise stated, we shall consider the designated set of the product algebra $\mathbb {A} \times \mathbb {B}$ as $D_{\mathbf {A}} \times D_{\mathbf {B}}$ , where $D_{\mathbf {A}}$ and $D_{\mathbf {B}}$ are the designated sets of $\mathbb {A}$ and $\mathbb {B}$ , respectively. Indeed, it is easy to check that $D_{\mathbf {A}} \times D_{\mathbf {B}}$ is a designated set of $\mathbb {A} \times \mathbb {B}$ . We denote this designated set $D_{\mathbf {A}} \times D_{\mathbf {B}}$ of $\mathbb {A} \times \mathbb {B}$ by $D_{\mathbb {A} \times \mathbb {B}}$ .

Since, the validity of a formula $\varphi $ of $\mathcal {L}_{\mathbb {A} \times \mathbb {B}}$ depends on whether the algebraic value of $[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {A} \times \mathbb {B}}$ , which is an element of $\mathbf {A} \times \mathbf {B}$ , belongs to the set $D_{\mathbf {A}} \times D_{\mathbf {B}}$ , an immediate question consists in asking whether the value of $[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {A} \times \mathbb {B}}$ depends on the values of $[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {A}}$ and $[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {B}}$ . If this is the case, then validity in $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ can be transferred to the validity in $\mathbf {V}^{(\mathbb {A})}$ and $\mathbf {V}^{(\mathbb {B})}$ separately. A positive answer of this question will be given in Theorem 3.17 In order to do so we need Definition 3.15 and Lemma 3.16 which will explain how the value of $[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {A} \times \mathbb {B}}$ can be calculated coordinate wise.

Then Theorem 3.17 is enough to show Theorem 3.19, which in turn states that if $\mathrm {BQ}_\varphi $ holds in both of $\mathbf {V}^{(\mathbb {A})}$ and $\mathbf {V}^{(\mathbb {B})}$ then $\mathrm {BQ}_\varphi $ also holds in $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ , for every formula $\varphi $ . Thus Theorem 3.19 represents the cornerstone for the constructions of product algebra-valued models of the negation free fragment of $\mathsf {ZF}$ .

We shall use (meta-)induction to prove the following results. This principle can be proved in $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ by using the same rank arguments used for Boolean-valued models [Reference Bell3, induction principle 1.7]. In $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ , the (meta-)induction principle states that, for every property $\Phi $ of names, if for all $u \in \mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ ,

$$ \begin{align*}&\forall v \in \mathrm{dom}(u) (\Phi(v)) \mbox{ implies } \Phi(u), \end{align*} $$

then every $u \in \mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ has the property $\Phi $ .

From the definition it is clear that for any $u \in \mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ , $\bar {u} \in \mathbf {V}^{(\mathbb {A})}$ and $\underline {u} \in \mathbf {V}^{(\mathbb {B})}$ .

Proof. (i) The proof is done by (meta-)induction. Let $v \in \mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ be an element such that for any $u \in \mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ and $w \in \mathrm {dom}(v)$ , we have

$$\begin{align*}&[\kern-1.5pt[ u = w ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}} = ([\kern-1.5pt[ \bar{u} = \bar{w} ]\kern-1.5pt]_{\mathbb{A}}, [\kern-1.5pt[ \underline{u} = \underline{w} ]\kern-1.5pt]_{\mathbb{B}}),\end{align*}$$

where suppose $\mathbb {A} = \langle \mathbf {A}, \wedge _{\mathbf {A}}, \vee _{\mathbf {A}}, \Rightarrow _{\mathbf {A}}, ^{*_{\mathbf {A}}}, \mathbf {1}_{\mathbf {A}}, \mathbf {0}_{\mathbf {A}} \rangle $ , $\mathbb {B} = \langle \mathbf {B}, \wedge _{\mathbf {B}}, \vee _{\mathbf {B}}, \Rightarrow _{\mathbf {B}}, ^{*_{\mathbf {B}}}, \mathbf {1}_{\mathbf {B}}, \mathbf {0}_{\mathbf {B}} \rangle $ and $\mathbb {A} \times \mathbb {B} = \langle \mathbf {A} \times \mathbf {B}, \wedge , \vee , \Rightarrow , ^*, \mathbf {1}, \mathbf {0} \rangle $ . It is then sufficient to prove that $[\kern-1.5pt[ u = v ]\kern-1.5pt] = ([\kern-1.5pt[ \bar {u} = \bar {v} ]\kern-1.5pt] , [\kern-1.5pt[ \underline {u} = \underline {v} ]\kern-1.5pt] )$ .

$$ \begin{align*} & [\kern-1.5pt[ u = v ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}} \\ & \quad= \bigwedge_{x \in \mathrm{dom}(u)}(u(x) \Rightarrow [\kern-1.5pt[ x \in v ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}}) \wedge \bigwedge_{w \in \mathrm{dom}(v)}(v(w) \Rightarrow [\kern-1.5pt[ w \in u ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}}), \mbox{ definition of } [\kern-1.5pt[ \cdot = \cdot ]\kern-1.5pt]\\ & \quad= \bigwedge_{x \in \mathrm{dom}(u)}(u(x) \Rightarrow \bigvee_{w \in \mathrm{dom}(v)} (v(w) \wedge [\kern-1.5pt[ w = x ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}})) \wedge \\ & \quad \bigwedge_{w \in \mathrm{dom}(v)}(v(w) \Rightarrow \bigvee_{x \in \mathrm{dom}(u)} (u(x) \wedge [\kern-1.5pt[ x = w ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}})), \mbox{ using the definition of } [\kern-1.5pt[ \cdot \in \cdot ]\kern-1.5pt]\\ &\quad = \bigwedge_{x \in \mathrm{dom}(u)}(\bar{u}(\bar{x}) \Rightarrow_{\mathbb{A}} \bigvee_{w \in \mathrm{dom}(v)} (\bar{v}(\bar{w}) \wedge_{\mathbb{A}} [\kern-1.5pt[ \bar{w} = \bar{x} ]\kern-1.5pt]_{\mathbb{A}}), \; \underline{u}(\underline{x})\Rightarrow_{\mathbb{B}}\\ &\quad \bigvee_{w \in \mathrm{dom}(v)} (\underline{v}(\underline{w}) \wedge_{\mathbb{B}} [\kern-1.5pt[ \underline{w} = \underline{x} ]\kern-1.5pt]_{\mathbb{B}})) \wedge\bigwedge_{w \in \mathrm{dom}(v)}(\bar{v}(\bar{w}) \Rightarrow_{\mathbb{A}}\\ & \quad \bigvee_{x \in \mathrm{dom}(u)} (\bar{u}(\bar{x}) \wedge_{\mathbb{A}} [\kern-1.5pt[ \bar{x} = \bar{w} ]\kern-1.5pt]_{\mathbb{A}}), \; \underline{v}(\underline{w})\Rightarrow_{\mathbb{B}} \bigvee_{x \in \mathrm{dom}(u)} (\underline{u}(\underline{x}) \wedge_{\mathbb{B}} [\kern-1.5pt[ \underline{x} = \underline{w} ]\kern-1.5pt]_{\mathbb{B}})),\\ & \mbox{using the induction hypothesis, that }u(x) = (\bar{u}(\bar{x}), \underline{u}(\underline{x})), \mbox{ and that }\\ & v(w) = (\bar{v}(\bar{w}), \underline{v}(\underline{w})), \mbox{ for all } x \in \mathrm{dom}(u) \mbox{ and } w \in \mathrm{dom}(v)\\ & \quad = ([\kern-1.5pt[ \bar{u} = \bar{v} ]\kern-1.5pt]_{\mathbb{A}}, \; [\kern-1.5pt[ \underline{u} = \underline{v} ]\kern-1.5pt]_{\mathbb{B}}), \mbox{ by using Theorem 3.11. and the definition of}\ [\kern-1.5pt[ \cdot = \cdot ]\kern-1.5pt]. \end{align*} $$

(ii) For any $u, v \in \mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$

$$ \begin{align*} [\kern-1.5pt[ u \in v ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}} & = \bigvee_{w \in \mathrm{dom}(v)}(v(w) \wedge [\kern-1.5pt[ w = u ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}})\\ & = \bigvee_{w \in \mathrm{dom}(v)}(v(w) \wedge ([\kern-1.5pt[ \bar{w} = \bar{u} ]\kern-1.5pt]_{\mathbb{A}}, [\kern-1.5pt[ \underline{w} = \underline{u} ]\kern-1.5pt]_{\mathbb{B}})), \mbox{ applying (i)}\\ & = (\bigvee_{\bar{w} \in \mathrm{dom}(\bar{v})} (\bar{v}(\bar{w}) \wedge_{\mathbb{A}} [\kern-1.5pt[ \bar{w} = \bar{u} ]\kern-1.5pt]_{\mathbb{A}}), \; \bigvee_{\underline{w} \in \mathrm{dom}(\underline{v})} (\underline{v}(\underline{w}) \wedge_{\mathbb{B}} [\kern-1.5pt[ \underline{w} = \underline{u} ]\kern-1.5pt]_{\mathbb{B}})), \\ & \mbox{ by Theorem 3.11. and the fact that}\ v(w) = (\bar{v}(\bar{w}), \underline{v}(\underline{w})),\\ &\mbox{ for all } w \in \mathrm{dom}(v)\\ & = ([\kern-1.5pt[ \bar{u} \in \bar{v} ]\kern-1.5pt]_{\mathbb{A}}, [\kern-1.5pt[ \underline{u} \in \underline{v} ]\kern-1.5pt]_{\mathbb{B}}). \end{align*} $$

As a conclusion of Lemma 3.16 we have the following theorem.

Theorem 3.17. Let $\mathbb {A}$ and $\mathbb {B}$ be two complete $\Lambda $ -algebras. If $\varphi (x_1, \ldots , x_n)$ is a formula of $\mathcal {L}_{\Lambda , \in }$ , the language of $\mathsf {ZFC}$ , having n free variables $x_1, \ldots , x_n$ , then for any $u_1, \ldots , u_n \in \mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ ,

$$ \begin{align*}[\kern-1.5pt[ \varphi(u_1, \ldots, u_n) ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}} = ([\kern-1.5pt[ \varphi(\bar{u_1}, \ldots, \bar{u_n}) ]\kern-1.5pt]_{\mathbb{A}}, [\kern-1.5pt[ \varphi(\underline{u_1}, \ldots, \underline{u_n}) ]\kern-1.5pt]_{\mathbb{B}}).\end{align*} $$

We now prove that not only the properties of the algebra but the property $\mathrm {BQ}_\varphi $ for any formula $\varphi $ is also hereditary in the product algebras.

Proof. Consider two complete $\Lambda $ -algebras $\mathbb {A} = \langle \mathbf {A}, \wedge _{\mathbf {A}}, \vee _{\mathbf {A}}, \Rightarrow _{\mathbf {A}}, ^{*_{\mathbf {A}}}, \mathbf {1}_{\mathbf {A}}, \mathbf {0}_{\mathbf {A}} \rangle $ and $\mathbb {B} = \langle \mathbf {B}, \wedge _{\mathbf {B}}, \vee _{\mathbf {B}}, \Rightarrow _{\mathbf {B}}, ^{*_{\mathbf {B}}}, \mathbf {1}_{\mathbf {B}}, \mathbf {0}_{\mathbf {B}} \rangle $ . Consider the product algebra $\mathbb {A} \times \mathbb {B}$ with the structure $\langle \mathbf {A} \times \mathbf {B}, \wedge , \vee , \Rightarrow , ^*, \mathbf {1}, \mathbf {0} \rangle $ . Let $\varphi $ be a formula such that $\mathrm {BQ}_\varphi $ holds in both $\mathbf {V}^{(\mathbb {A})}$ and $\mathbf {V}^{(\mathbb {B})}$ and let $u \in \mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ be any element. Then we have the following.

$$ \begin{align*} [\kern-1.5pt[\forall x & (x\in u \to \varphi(x))]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}} \\ & = ([\kern-1.5pt[\forall x(x\in \bar{u} \to \varphi(x))]\kern-1.5pt]_{\mathbb{A}}, \; [\kern-1.5pt[\forall x(x\in \underline{u} \to \varphi(x))]\kern-1.5pt]_{\mathbb{B}}), \mbox{ by Theorem 3.17.}\\ & = (\bigwedge_{x\in\mathrm{dom}(\bar{u})} (\bar{u}(x)\Rightarrow_{\mathbb{A}} [\kern-1.5pt[\varphi(x)]\kern-1.5pt]_{\mathbb{A}}), \; \bigwedge_{x\in\mathrm{dom}(\underline{u})} (\underline{u}(x)\Rightarrow_{\mathbb{B}} [\kern-1.5pt[\varphi(x)]\kern-1.5pt]_{\mathbb{B}})),\\ & \mbox{ since}\ \mathrm{BQ}_\varphi\ \mbox{holds in}\ \mathbf{V}^{(\mathbb{A})}\ \mbox{and}\ \mathbf{V}^{(\mathbb{B})}\ \mbox{both}\\ & = (\bigwedge_{x\in\mathrm{dom}(u)} (\bar{u}(\bar{x})\Rightarrow_{\mathbb{A}} [\kern-1.5pt[\varphi(\bar{x})]\kern-1.5pt]_{\mathbb{A}}), \; \bigwedge_{x\in\mathrm{dom}(u)} (\underline{u}(\underline{x})\Rightarrow_{\mathbb{B}} [\kern-1.5pt[\varphi(\underline{x})]\kern-1.5pt]_{\mathbb{B}})),\\ & \mbox{ follows from the definitions of}\ \bar{u}\ \mbox{and}\ \underline{u}\\ & = \bigwedge_{x\in\mathrm{dom}(u)} ((\bar{u}(\bar{x})\Rightarrow_{\mathbb{A}} [\kern-1.5pt[\varphi(\bar{x})]\kern-1.5pt]_{\mathbb{A}}),(\underline{u}(\underline{x})\Rightarrow_{\mathbb{B}} [\kern-1.5pt[\varphi(\underline{x})]\kern-1.5pt]_{\mathbb{B}})), \mbox{ by Theorem 3.11.(ii).}\\ & = \bigwedge_{x\in\mathrm{dom}(u)} ((\bar{u}(\bar{x}), \underline{u}(\underline{x})) \Rightarrow ([\kern-1.5pt[\varphi(\bar{x})]\kern-1.5pt]_{\mathbb{A}}, [\kern-1.5pt[\varphi(\underline{x})]\kern-1.5pt]_{\mathbb{B}})), \mbox{ by the definition of}\ \Rightarrow\\ & = \bigwedge_{x\in\mathrm{dom}(u)} (u(x)\Rightarrow [\kern-1.5pt[\varphi(x)]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}}),\\ & \mbox{by Theorem 3.17. and the definitions of}\ \bar{u}\ \mbox{and}\ \underline{u}. \end{align*} $$

Hence, $\mathrm {BQ}_\varphi $ holds in $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ .□

The following theorem acts as a backbone of the model constructions of non-classical set theories in this paper.

Definition 3.21. For a $\Lambda $ -algebra $\mathbb {A}$ with a designated set D, $\mathrm {ValSent}_{(\mathbb {A}, D)}$ is the collection of all sentences valid in $\mathbf {V}^{(\mathbb {A})}$ , i.e.,

$$\begin{align*}\mathrm{ValSent}_{(\mathbb{A},D)} = \{\varphi \in \mathrm{Sent}_{\Lambda, \in} : \mathbf{V}^{(\mathbb{A})} \models_D \varphi\}.\end{align*}$$

To keep the notation uniform with the other notations used in this paper, we sometimes denote $\mathrm {ValSent}_{(\mathbb {A}, D)}$ by $\mathrm {ValSent}_{\mathbb {A}}$ when the designated set D is clear from the context.

Proof. For any two complete $\Lambda $ -algebras $\mathbb {A}$ and $\mathbb {B}$ ,

$$ \begin{align*} \mathrm{ValSent}_{(\mathbb{A} \times \mathbb{B}, D_{\mathbb{A} \times \mathbb{B}})} & = \{\varphi \in \mathrm{Sent}_{\Lambda, \in} : \mathbf{V}^{(\mathbb{A} \times \mathbb{B})} \models_{D_{\mathbb{A} \times \mathbb{B}}} \varphi\}\\ & = \{\varphi \in \mathrm{Sent}_{\Lambda, \in} : [\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}} \in D_{\mathbb{A}} \times D_{\mathbb{B}}\},\\ & = \{\varphi \in \mathrm{Sent}_{\Lambda, \in} : ([\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{A}}, [\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{B}}) \in D_{\mathbb{A}} \times D_{\mathbb{B}}\}, \mbox{ by Corollary 3.18.} \\ & = \{\varphi \in \mathrm{Sent}_{\Lambda, \in} : [\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{A}} \in D_{\mathbb{A}}\} \cap \{\varphi \in \mathrm{Sent}_{\Lambda, \in} : [\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{B}} \in D_{\mathbb{B}}\} \\ & = \mathrm{ValSent}_{(\mathbb{A}, D_{\mathbb{A}})} \cap \mathrm{ValSent}_{(\mathbb{B}, D_{\mathbb{B}})}. \end{align*} $$

This completes the proof.□

Notice that Observation 3.22 depends on the specific choice of the designated set of $\mathbb {A} \times \mathbb {B}$ , that is $D_{\mathbb {A} \times \mathbb {B}}$ , in terms of the product of the single designated sets $D_{\mathbb {A}}$ and $ D_{\mathbb {B}}$ . To see this, consider the case when both the algebras $\mathbb {A}$ and $\mathbb {B}$ are equal to $\mathrm {PS}_3$ . The designated set of $\mathrm {PS}_3$ is

. Let us now take the following formula in $\mathrm {Sent}_{\Lambda , \in }$ :

(Par) $$ \begin{align} \exists x \exists y \exists z (z \in x \wedge z \notin y \wedge x = y). \end{align} $$

It was proved in [Reference Löwe and Tarafder13, theorem 6.2] that

, i.e., $\mathbf {V}^{(\mathrm {PS}_3)} \models _{D_{\mathrm {PS}_3}} \mathsf {Par}$ . Hence, $\mathsf {Par} \in \mathrm {ValSent}_{(\mathrm {PS}_3, D_{\mathrm {PS}_3})} \cap \mathrm {ValSent}_{(\mathrm {PS}_3, D_{\mathrm {PS}_3})}$ . If the designated set of the product algebra $\mathrm {PS}_3 \times \mathrm {PS}_3$ was taken to be $D = \{(\mathbf {1}, \mathbf {1})\}$ , instead of $D_{\mathrm {PS}_3 \times \mathrm {PS}_3}$ , then

Hence, by the definition of validity,

$$\begin{align*}\mathbf{V}^{(\mathrm{PS}_3 \times \mathrm{PS}_3)} \not\models_D \mathsf{Par}, \mbox{ i.e., } \mathsf{Par} \notin \mathrm{ValSent}_{(\mathrm{PS}_3 \times \mathrm{PS}_3, D)},\end{align*}$$

which implies that $\mathrm {ValSent}_{(\mathrm {PS}_3 \times \mathrm {PS}_3, D)} \subsetneqq \mathrm {ValSent}_{(\mathrm {PS}_3, D_{\mathrm {PS}_3})} \cap \mathrm {ValSent}_{(\mathrm {PS}_3, D_{\mathrm {PS}_3})}$ .

3.3 Invalidity in product algebras

So far we discussed validity in product algebra-valued models, showing that the product structure determines a coordinate-wise notion of validity. This is enough to transfer from the single algebra-valued models to their product the validity of the negation free fragment of $\mathsf {ZF}$ .

Since the general goal of this work is to discuss independence in non-classical set theory, we also need to discuss the notion of invalidity. In this respect, we notice that the coordinate-wise functioning of validity generates a fundamental mismatch between the invalidity of a formula and the validity of its negation. Indeed, given a formula $\varphi \in \mathrm {Sent}_{\Lambda , \in }$ , we have that $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})} \models \varphi $ iff $[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {A} \times \mathbb {B}} \in D_{\mathbb {A} \times \mathbb {B}}$ iff $\big {(}[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {A}} \in D_{\mathbb {A}}$ and $[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {B}} \in D_{\mathbb {B}}\big {)}$ . Therefore, we get that $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})} \not \models \varphi $ iff $\big {(}[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {A}} \notin D_{\mathbb {A}}$ or $[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {B}} \notin D_{\mathbb {B}}\big {)}$ . However, this does not necessarily means that $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})} \models \neg \varphi $ , since this holds only when $\big {(}[\kern-1.5pt[ \neg \varphi ]\kern-1.5pt] _{\mathbb {A}} \in D_{\mathbb {A}}$ and $[\kern-1.5pt[ \neg \varphi ]\kern-1.5pt] _{\mathbb {B}} \in D_{\mathbb {B}}\big {)}$ .

Let us consider a concrete case, within a Boolean setting. Consider two complete Boolean algebras $\mathbb {B}_1$ and $\mathbb {B}_2$ and two ultrafilters, $D_{\mathbb {B}_1}$ and $D_{\mathbb {B}_2}$ of $\mathbb {B}_1$ and $\mathbb {B}_2$ , respectively, such that the corresponding Boolean-valued models validate, respectively, $\mathsf {CH}$ and $\neg \mathsf {CH}$ , say, $\mathbf {V}^{(\mathbb {B}_1)} \models _{D_{\mathbb {B}_1}}\mathsf {CH}$ and $\mathbf {V}^{(\mathbb {B}_2)} \models _{D_{\mathbb {B}_2}} \neg \mathsf {CH}$ . First of all notice that, the product algebra $\mathbb {B}_1 \times \mathbb {B}_2$ is a Boolean algebra, as the operations are defined coordinate-wise. Now, because of the validity in $\mathbf {V}^{(\mathbb {B}_2)}$ , we have that $\mathbf {V}^{(\mathbb {B}_1 \times \mathbb {B}_2)} \not \models _{D_{\mathbb {B}_1 \times \mathbb {B}_2}} \mathsf {CH}$ . However, because of the validity in $\mathbf {V}^{(\mathbb {B}_1)}$ , we also have that $\mathbf {V}^{(\mathbb {B}_1 \times \mathbb {B}_2)} \not \models _{D_{\mathbb {B}_1 \times \mathbb {B}_2}} \neg \mathsf {CH}$ . This observation is even more striking if we realize that, since both $\mathbf {V}^{(\mathbb {B}_1)}$ and $\mathbf {V}^{(\mathbb {B}_2)}$ are classical models of $\mathsf {ZF}$ , we have that classical logic, including Terzium non Datur, is valid in $\mathbf {V}^{(\mathbb {B}_1 \times \mathbb {B}_2)}$ . Hence, for every formula $\varphi $ we have $\mathbf {V}^{(\mathbb {B}_1 \times \mathbb {B}_2)} \models _{D_{\mathbb {B}_1 \times \mathbb {B}_2}} \varphi \lor \neg \varphi $ . Hence, in particular, $\mathbf {V}^{(\mathbb {B}_1 \times \mathbb {B}_2)} \models _{D_{\mathbb {B}_1 \times \mathbb {B}_2}} \mathsf {CH} \lor \neg \mathsf {CH}$ .

The explanation for this peculiar phenomenon is twofold. On the one hand we can simply notice that the filter $D_1 \times D_2$ is not an ultrafilter (although both $D_1$ and $D_2$ are). On the other hand, we can also notice that it is the peculiar structure of the product algebra-valued models which is responsible for the indeterminateness of $\mathsf {CH}$ . Indeed, it is exactly the use of a coordinate-wise notion of validity that allows these structures to internalize the meta-theoretical indeterminacy of the truth-value of a sentence like $\mathsf {CH}$ . For this same reason, these structures seem perfectly suited to provide a fine-grained analysis of independence in set theory.Footnote ⁷ Toward this goal let us define what we mean by independence in this context.

The choice of the following definition is motivated by the attempt to separate the notion of independence from the specific (and possibly peculiar) properties of negation. Moreover, it is classically equivalent to the standard one.

Thanks to Definition 3.23 we can account for proper cases of independence, even in the context of paraconsistent negations. For example, in the case of $\mathrm {PS}_3$ , we have that if , then . Thus, every sentence receiving the intermediate value of $\mathrm {PS}_3$ (and showing the paraconsistency of $\mathbf {V}^{(\mathrm {PS}_3)}$ ) could automatically be understood as independent from the set theory of $\mathbf {V}^{(\mathrm {PS}_3)}$ . Thus Definition 3.23 avoids these trivial cases allowing one to account for real instances of independence in non-classical set theories.

Before using product algebras to provide independence results, we will devote an entire section to the study of the different logics that can result by taking products of well-known algebras. To this end, we will revise few definitions from the literature. These are devised to account for the variability that can subsist between the logic associate to an algebra and the one associated to an algebra-valued model built from that algebra [Reference Löwe, Paßmann and Tarafder12].

4.1 Loyalty and faithfulness with respect to product algebras

In this section we will follow the notations of [Reference Löwe, Paßmann and Tarafder12]. For any two structures $\mathcal {U}_1 = \langle \mathbf {U}_1, \Lambda \rangle $ and $\mathcal {U}_2 = \langle \mathbf {U}_2, \Lambda \rangle $ having domains $\mathbf {U}_1$ and $\mathbf {U}_2$ , respectively, and operations corresponding to all the connectives in $\Lambda $ , a map $f : \mathbf {U}_1 \to \mathbf {U}_2$ is said to be a $\Lambda $ -homomorphism if it preserves all the connectives in $\Lambda $ . A $\Lambda $ -homomorphism f is said to be a $\Lambda $ -isomorphism if in addition f is a bijective function. A $\Lambda $ -isomorphism from a structure into itself is said to be a $\Lambda $ -automorphism.

In Section 2.1, we defined that $\mathcal {L}_\Lambda $ is the collection of all propositional formulas and $\mathrm {Sent}_{\Lambda ,\in }$ is the collection of all sentences in $\mathcal {L}_{\Lambda , \in }$ . Let us now consider any $\Lambda $ -algebra $\mathbb {A}$ , having domain $\mathbf {A}$ . Then, for the structures $\langle \mathcal {L}_\Lambda , \Lambda \rangle $ and $\mathbb {A}$ , any $\Lambda $ -homomorphisms $v: \mathcal {L}_\Lambda \to \mathbf {A}$ are called $\mathbb {A}$ -assignments. Similarly, consider the two structures $\langle \mathcal {L}_\Lambda , \Lambda \rangle $ and $\langle \mathrm {Sent}_{\Lambda , \in }, \Lambda \rangle $ ; $\Lambda $ -homomorphisms $T:\mathcal {L}_\Lambda \to \mathrm {Sent}_{\Lambda ,\in }$ are called translations.

For a $\Lambda $ -algebra $\mathbb {A}$ and a designated set D of $\mathbb {A}$ , following the standard way, the propositional logic of $(\mathbb {A},D)$ is defined as

$$ \begin{align*}\mathbf{L}(\mathbb{A},D) := \{\varphi\in\mathcal{L}_\Lambda\,:\,v(\varphi)\in D\mbox{ for all}\ \mathbb{A}\mbox{-assignments}\ v\}.\end{align*} $$

Notice that if $\mathbb {B}$ is a Boolean algebra and D is any filter, then $\mathbf {L}(\mathbb {B},D) = \mathrm {CPL}$ , the classical propositional logic.

Let $\mathbb {A}$ be a $\Lambda $ -algebra having domain $\mathbf {A}$ and the designated set D. Corresponding to the two structures $\langle \mathrm {Sent}_{\Lambda , \in }, \Lambda \rangle $ and $\mathbb {A}$ , the map $[\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}}$ restricted over $\mathrm {Sent}_{\Lambda , \in }$ , $[\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}} \! \restriction _{\mathrm {Sent}_{\Lambda ,\in }}:\mathrm {Sent}_{\Lambda ,\in }\to \mathbf {A}$ is a $\Lambda $ -homomorphism. Following [Reference Löwe, Paßmann and Tarafder12, sec. 2.6], we will define the propositional logic of $([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}} \! \restriction _{\mathrm {Sent}_{\Lambda ,\in }},D)$ as

$$ \begin{align*}\mathbf{L}([\kern-1.5pt[\cdot]\kern-1.5pt]_{\mathbb{A}} \! \restriction_{\mathrm{Sent}_{\Lambda,\in}},D) := \{\varphi\in\mathcal{L}_\Lambda\,:\,[\kern-1.5pt[ T(\varphi)]\kern-1.5pt]\in D\mbox{ for all translations}\ T\}.\end{align*} $$

To make the notation more readable, from now on, we will denote $\mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}} \! \restriction _{\mathrm {Sent}_{\Lambda ,\in }},D)$ by $\mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}},D)$ . Note that, the collection $\mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}},D)$ contains all those propositional formulas $\varphi $ such that if every propositional variable of $\varphi $ is replaced by an arbitrarily chosen set theoretic sentence then the resultant sentence remains valid in $\mathbf {V}^{(\mathbb {A})}$ . Intuitively, $\mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}},D)$ is the logic of the algebra-valued model $\mathbf {V}^{(\mathbb {A})}$ .

It is not hard to check that, for any $\Lambda $ -algebra $\mathbb {A}$ and a designated set D we have, $\mathbf {L}(\mathbb {A},D) \subseteq \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}},D)$ .

Intuitively, the loyalty confirms that the logic of an algebra $\mathbb {A}$ and the logic of its corresponding algebra-valued model coincide. Hence, abusing notation, sometimes we shall refer to the fact that $[\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}}\!\restriction _{\mathrm {Sent}_{\Lambda ,\in }}$ is loyal to $(\mathbb {A}, D)$ by saying that $\mathbf {V}^{(\mathbb {A})}$ is loyal to $(\mathbb {A}, D)$ .

Although counter-intuitive, it is not the case that for any algebra $\mathbb {A}$ the algebra-valued model $\mathbf {V}^{(\mathbb {A})}$ is loyal to $(\mathbb {A}, D)$ and already at the level of Heyting algebras we find cases of illoyal structures in [Reference Löwe, Paßmann and Tarafder12, sec. 5.2]. For example, consider the Heyting algebra $\mathbb {H}_5$ (displayed in Figure 1) of five elements which is the tail stretch of the four-valued Boolean algebra $\mathbb {B}_4 = \langle \mathbf {B}, \wedge _{\mathbf {B}}, \vee _{\mathbf {B}}, \Rightarrow _{\mathbf {B}}, ^{*_{\mathbf {B}}}, \mathbf {1}, \mathbf {0} \rangle $ (displayed in Figure 2) by adding one element $\mathbf {1}'$ at the top of $\mathbb {B}_4$ , where .

The structure $\mathbb {H}_5$ becomes a complete $\Lambda $ -algebra having $\mathbf {1}'$ as the top element and $\mathbb {B}_4$ as a substructure, where the operator $\Rightarrow $ of $\mathbb {H}_5$ is defined as follows:

$$ \begin{align*}a \Rightarrow b := \left\{ \begin{array}{cl} a \Rightarrow_{\mathbf{B}} b & \mbox{if}\ a,b\in \mathbf{B}\ \mbox{such that}\ a\not\leq b,\\ \mathbf{1}' & \mbox{if}\ a,b\in \mathbf{B}\ \mbox{with}\ a\leq b\ \mbox{or if}\ b=\mathbf{1}',\\ b & \mbox{if}\ a=\mathbf{1}',\\ \end{array}\right.\end{align*} $$

One can check that $\mathbb {H}_5$ is a Heyting algebra and that $\mathbf {V}^{(\mathbb {H}_5)}$ is illoyal to $(\mathbb {H}_5, \{\mathbf {1}'\})$ .

Figure 1 The Heyting algebra $\mathbb {H}_5$ .

Figure 2 The Boolean algebra $\mathbb {B}_4$ .

To unify the notations of loyalty and faithfulness, the fact that $[\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}}\!\restriction _{\mathrm {Sent}_{\Lambda ,\in }}$ is faithful to $\mathbb {A}$ will be expressed by saying that $\mathbf {V}^{(\mathbb {A})}$ is faithful to $\mathbb {A}$ . Observe that the notion of faithfulness is independent of the choice of the designated set of $\mathbb {A}$ .

We shall now explore the loyalty and faithfulness of the product-algebra-valued models depending on the loyalty and faithfulness of the algebra-valued models of the component algebras. In this process we need the following theorem.

4.1.1 Product of algebras corresponding to two loyal models

As expected, the logic of the product-algebra-valued model will be equal to the intersection of the logics of the algebra-valued models of the component algebras.

Theorem 4.28. For any two $\Lambda $ -algebras $\mathbb {A}$ and $\mathbb {B}$ having the designated sets $D_{\mathbb {A}}$ and $D_{\mathbb {B}}$ , respectively, $\mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A} \times \mathbb {B}}, D_{\mathbb {A} \times \mathbb {B}}) = \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}}, D_{\mathbb {A}}) \cap \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {B}}, D_{\mathbb {B}})$ .

Proof. For a formula $\varphi \in \mathcal {L}_\Lambda $ , $\varphi \notin \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A}}, D_{\mathbb {A}}) \cap \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {B}}, D_{\mathbb {B}})$ iff there exists a translation $T:\mathcal {L}_\Lambda \to \mathrm {Sent}_{\Lambda ,\in }$ such that either $[\kern-1.5pt[ T(\varphi ) ]\kern-1.5pt] _{\mathbb {A}} \notin D_{\mathbb {A}}$ or $[\kern-1.5pt[ T(\varphi ) ]\kern-1.5pt] _{\mathbb {B}} \notin D_{\mathbb {B}}$ or both iff $([\kern-1.5pt[ T(\varphi ) ]\kern-1.5pt] _{\mathbb {A}}, [\kern-1.5pt[ T(\varphi ) ]\kern-1.5pt] _{\mathbb {B}}) \notin D_{\mathbb {A} \times \mathbb {B}}$ iff $[\kern-1.5pt[ T(\varphi ) ]\kern-1.5pt] _{\mathbb {A} \times \mathbb {B}} \notin D_{\mathbb {A} \times \mathbb {B}}$ , by Corollary 3.18 iff $\varphi \notin \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {A} \times \mathbb {B}}, D_{\mathbb {A} \times \mathbb {B}})$ .□

Theorem 4.29. Let $\mathbb {A}$ and $\mathbb {B}$ be two $\Lambda $ -algebras having the designated sets $D_{\mathbb {A}}$ and $D_{\mathbb {B}}$ , respectively. If $\mathbf {V}^{(\mathbb {A})}$ and $\mathbf {V}^{(\mathbb {B})}$ are loyal to $(\mathbb {A}, D_{\mathbb {A}})$ and $(\mathbb {B}, D_{\mathbb {B}})$ , respectively, then $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ is loyal to $(\mathbb {A} \times \mathbb {B}, D_{\mathbb {A} \times \mathbb {B}})$ .

Proof. We have

$$ \begin{align*} \mathbf{L}([\kern-1.5pt[\cdot]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}}, D_{\mathbb{A} \times \mathbb{B}}) & = \mathbf{L}([\kern-1.5pt[\cdot]\kern-1.5pt]_{\mathbb{A}}, D_{\mathbb{A}}) \cap \mathbf{L}([\kern-1.5pt[\cdot]\kern-1.5pt]_{\mathbb{B}}, D_{\mathbb{B}}), \mbox{ by Theorem 4.28.}\\ & = \mathbf{L}(\mathbb{A}, D_{\mathbb{A}}) \cap \mathbf{L}(\mathbb{B}, D_{\mathbb{B}}), \mbox{ since}\ \mathbf{V}^{(\mathbb{A})}\ \mbox{and} \mathbf{V}^{(\mathbb{B})}\ \mbox{are loyal to}\ (\mathbb{A}, D_{\mathbb{A}})\\ & \mbox{ and}\ (\mathbb{B}, D_{\mathbb{B}}),\ \mbox{respectively}\\ & = \mathbf{L}(\mathbb{A} \times \mathbb{B}, D_{\mathbb{A} \times \mathbb{B}}), \mbox{ by the definition of the product algebra}. \end{align*} $$

This completes the proof.□

4.1.2 Product of algebras corresponding to one loyal and one illoyal model

First we shall give examples of two $\Lambda $ -algebras $\mathbb {A}$ and $\mathbb {B}$ such that $\mathbf {V}^{(\mathbb {A})}$ is loyal to $(\mathbb {A}, D_{\mathbb {A}})$ , $\mathbf {V}^{(\mathbb {B})}$ is illoyal to $(\mathbb {B}, D_{\mathbb {B}})$ , and $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ is illoyal to $(\mathbb {A} \times \mathbb {B}, D_{\mathbb {A} \times \mathbb {B}})$ .

Let us first consider the two-valued Boolean algebra $\mathbb {B}_2$ . Notice that $\mathbf {V}^{(\mathbb {B}_2)}$ is loyal to $(\mathbb {B}_2, \{\mathbf {1}\})$ since we have that $\mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {B}_2}, \{\mathbf {1}\}) = \mathrm {CPL} = \mathbf {L}(\mathbb {B}_2, \{\mathbf {1}\})$ , where $\mathbf {1}$ is assumed to be the top element of $\mathbb {B}_2$ .

Second, consider a four-valued algebra such that the operators $\wedge , \vee ,$ and $\Rightarrow $ are exactly as those of the four-valued Boolean algebra $\mathbb {B}_4$ , where $\mathbf {1}$ and $\mathbf {0}$ are the top and bottom elements of the lattice, respectively, and are the two intermediate incomparable values. The unary operator $^*$ of $\mathbb {BH}$ is defined as follows: and $\mathbf {0}^* = \mathbf {1}$ . Observe that, there exists a non-trivial automorphism $f : \mathbb {BH} \to \mathbb {BH}$ , defined as $f(\mathbf {1}) = \mathbf {1}$ , $f(\mathbf {0}) = \mathbf {0}$ , , and . Hence, there is no $\varphi \in \mathrm {Sent}_{\Lambda , \in }$ such that or , by Theorem 4.27 This leads to the fact that the range of $[\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {BH}}$ is $\mathbb {B}_2$ and hence $\mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {BH}}, \{\mathbf {1}\}) = \mathrm {CPL}$ . But for $\varphi \in \mathcal {L}_\Lambda $ , the formula $\varphi \vee \lnot \varphi \notin \mathbf {L}(\mathbb {BH}, \{\mathbf {1}\})$ , i.e., $\mathbf {L}(\mathbb {BH}, \{\mathbf {1}\}) \subsetneqq \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {BH}}, \{\mathbf {1}\})$ . Therefore $\mathbf {V}^{(\mathbb {BH})}$ is illoyal to $(\mathbb {BH}, \{\mathbf {1}\})$ .

By Theorem 4.28, $\mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {B}_2 \times \mathbb {BH}}, \{(\mathbf {1}, \mathbf {1})\}) = \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {B}_2}, \{\mathbf {1}\}) \; \cap \; \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {BH}}, \{\mathbf {1}\}) = \mathrm {CPL}$ . But $\mathbf {L}(\mathbb {B}_2 \times \mathbb {BH}, \{(\mathbf {1}, \mathbf {1}\}) \neq \mathrm {CPL}$ as $\varphi \vee \lnot \varphi \notin \mathbf {L}(\mathbb {B}_2 \times \mathbb {BH}, \{(\mathbf {1}, \mathbf {1})\})$ .

Hence, we get that $\mathbf {V}^{(\mathbb {B}_2 \times \mathbb {BH})}$ is illoyal to $(\mathbb {B}_2 \times \mathbb {BH}, \{(\mathbf {1}, \mathbf {1})\})$ .

Question 4.1.3. Do there exist $\Lambda $ -algebras $\mathbb {A}$ and $\mathbb {B}$ such that $\mathbf {V}^{(\mathbb {A})}$ is loyal to $(\mathbb {A}, D_{\mathbb {A}})$ , $\mathbf {V}^{(\mathbb {B})}$ is illoyal to $(\mathbb {B}, D_{\mathbb {B}})$ , and $\mathbf {V}^{(\mathbb {A} \times \mathbb {B})}$ is loyal to $(\mathbb {A} \times \mathbb {B}, D_{\mathbb {A} \times \mathbb {B}})$ ?

We shall give a partial answer to Question 4.1.3 If there exists a Heyting algebra $\mathbb {H}$ such that the intuitionistic propositional logic $\mathrm {IPL}$ is complete with respect to $\mathbb {H}$ and $\mathbf {V}^{(\mathbb {H})}$ is loyal to $(\mathbb {H}, \{\mathbf {1}\})$ then the answer to Question 4.1.3 will be affirmative. For such an algebra $\mathbb {H}$ , if exists, we have $\mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {H}}, \{\mathbf {1}\}) = \mathbf {L}(\mathbb {H}, \{\mathbf {1}\}) = \mathrm {IPL}$ . Let us one more time consider the Heyting algebra $\mathbb {H}_5$ and the $\Lambda $ -automorphism $f : \mathbb {H}_5 \to \mathbb {H}_5$ , defined as $f(\mathbf {1}') = \mathbf {1}', f(\mathbf {1}) = \mathbf {1}$ , $f(\mathbf {0}) = \mathbf {0}$ , , and . By Theorem 4.27, we can conclude that the range of $[\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {H}_5}\!\restriction \mathrm {Sent}_{\Lambda , \in }$ contains neither nor , which produces the three-valued Heyting algebra $\mathbb {H}_3$ . Hence $\mathrm {IPL} \subseteq \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {H}_5}, \{\mathbf {1}'\})$ . By our assumption $\mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {H}}, \{\mathbf {1}\}) = \mathbf {L}(\mathbb {H}, \{\mathbf {1}\}) = \mathrm {IPL}$ . So we get

$$ \begin{align*} \mathbf{L}([\kern-1.5pt[\cdot]\kern-1.5pt]_{\mathbb{H} \times \mathbb{H}_5}, \{(\mathbf{1}, \mathbf{1}')\}) & = \mathbf{L}([\kern-1.5pt[\cdot]\kern-1.5pt]_{\mathbb{H}}, \{(\mathbf{1})\}) \; \cap \; \mathbf{L}([\kern-1.5pt[\cdot]\kern-1.5pt]_{\mathbb{H}_5}, \{\mathbf{1}'\})\\ & = \mathrm{IPL}\\ & = \mathbf{L}(\mathbb{H}, \{(\mathbf{1})\}) \; \cap \; \mathbf{L}(\mathbb{H}_5, \{(\mathbf{1}')\})\\ & = \mathbf{L}(\mathbb{H} \times \mathbb{H}_5, \{(\mathbf{1}, \mathbf{1}')\}). \end{align*} $$

Note that if there exists an illoyal Boolean algebra $\mathbb {B}$ then also we get a positive answer to Question 4.1.3 by replacing $\mathbb {H}$ and $\mathbb {H}_5$ by $\mathbb {B}_2$ and $\mathbb {B}$ , respectively, in the above argument.

If such Boolean algebra $\mathbb {B}$ and Heyting algebra $\mathbb {H}$ exist then in addition we shall get $\mathbf {V}^{(\mathbb {H} \times \mathbb {H}_5)} \models \mathrm {IZF}$ and $\mathbf {V}^{(\mathbb {B}_2 \times \mathbb {B})} \models \mathrm {ZFC}$ . The reason being that for any axiom $\varphi $ of $\mathrm {IZF}$ , $\mathbf {V}^{(\mathbb {H})} \models \varphi $ and $\mathbf {V}^{(\mathbb {H}_5)} \models \varphi $ . Hence, $\varphi $ belongs to both $\mathrm {ValSent}_{\mathbb {H}}$ and $\mathrm {ValSent}_{\mathbb {H}_5}$ . This implies that $\varphi \in \mathrm {ValSent}_{\mathbb {H}} \cap \mathrm {ValSent}_{\mathbb {H}_5} = \mathrm {ValSent}_{\mathbb {H} \times \mathbb {H}_5}$ , by Observation 3.22 Hence, $\mathbf {V}^{(\mathbb {H} \times \mathbb {H}_5)} \models \varphi $ . Similarly, we can show that $\mathbf {V}^{(\mathbb {B}_2 \times \mathbb {B})} \models \mathrm {ZFC}$ .

4.2 The product algebra of a paraconsistent and a Heyting algebra

Notice that $\mathrm {PS}_3$ and $\mathbb {H}_3$ are two $\Lambda $ -algebras, having the same underlying set , where the designated sets corresponding to $\mathrm {PS}_3$ and $\mathbb {H}_3$ are and $D_{\mathbb {H}_3} = \{\mathbf {1}\}$ , respectively. We shall explore the product algebra $\mathrm {PS}_3 \times \mathbb {H}_3$ and its corresponding algebra-valued model. It is proved in [Reference Löwe, Paßmann and Tarafder12] that $\mathbf {V}^{(\mathrm {PS}_3)}$ is faithful to $\mathrm {PS}_3$ . On the other hand, $\mathbf {V}^{(\mathbb {H}_3)}$ is also faithful to $\mathbb {H}_3$ :

1. (i) $[\kern-1.5pt[ \forall x (x = x) ]\kern-1.5pt] _{\mathbb {H}_3} = \mathbf {1}$ ,

2. (ii) $[\kern-1.5pt[ \forall x (x \neq x) ]\kern-1.5pt] _{\mathbb {H}_3} = \mathbf {0}$ , and

3. (iii) .

Hence, using Theorem 4.26, one can get the following theorem.

The designated set $D_{\mathrm {PS}_3 \times \mathbb {H}_3}$ of $\mathrm {PS}_3 \times \mathbb {H}_3$ is . By Theorem 2.6, $\mathbf {L}(\mathrm {PS}_3, D_{\mathrm {PS}_3}) = \mathbb {L}\mathrm {PS}_3 \subsetneqq \mathrm {CPL}$ and $\mathbf {L}(\mathrm {PS}_3, D_{\mathrm {PS}_3})$ is paraconsistent. Similarly, $\mathbf {L}(\mathbb {H}_3, D_{\mathbb {H}_3}) = \mathrm {IPL} \subsetneqq \mathrm {CPL}$ .

Proof. It is easy to check that $\mathbf {L}(\mathrm {PS}_3 \times \mathbb {H}_3, D_{\mathrm {PS}_3 \times \mathbb {H}_3}) = \mathbf {L}(\mathrm {PS}_3, D_{\mathrm {PS}_3}) \cap \mathbf {L}(\mathbb {H}_3, D_{\mathbb {H}_3})$ . This entails that $\mathbf {L}(\mathrm {PS}_3 \times \mathbb {H}_3, D_{\mathrm {PS}_3 \times \mathbb {H}_3})$ is not $\mathrm {CPL}$ .

Since, there exist formulas $\varphi , \psi \in \mathcal {L}_\Lambda $ such that $(\varphi \wedge \lnot \varphi ) \to \psi \notin \mathbf {L}(\mathrm {PS}_3, D_{\mathrm {PS}_3})$ and $\varphi \vee \lnot \varphi \notin \mathbf {L}(\mathbb {H}_3, D_{\mathbb {H}_3})$ we get that

$$ \begin{align*}(\varphi \wedge \lnot \varphi) \to \psi, \; \varphi \vee \lnot \varphi \notin \mathbf{L}(\mathrm{PS}_3 \times \mathbb{H}_3, D_{\mathrm{PS}_3 \times \mathbb{H}_3})\end{align*} $$

as well. Hence, $\mathbf {L}(\mathrm {PS}_3 \times \mathbb {H}_3, D_{\mathrm {PS}_3 \times \mathbb {H}_3})$ is both paraconsistent and paracomplete.

On the other hand, we know that for any two formulas $\varphi , \psi \in \mathcal {L}_\Lambda $ , $(\varphi \wedge \lnot \varphi ) \to \psi $ is a theorem of $\mathrm {IPL}$ . Since there exist $\varphi , \psi \in \mathcal {L}_\Lambda $ such that $(\varphi \wedge \lnot \varphi ) \to \psi \notin \mathbf {L}(\mathrm {PS}_3 \times \mathbb {H}_3, D_{\mathrm {PS}_3 \times \mathbb {H}_3})$ , therefore $\mathbf {L}(\mathrm {PS}_3 \times \mathbb {H}_3, D_{\mathrm {PS}_3 \times \mathbb {H}_3})$ is not $\mathrm {IPL}$ .□

Hence we can conclude that $\mathbf {L}(\mathrm {PS}_3 \times \mathbb {H}_3, D)$ is a paraconsistent logic which is a proper subclass of $\mathbf {L}(\mathrm {PS}_3, D_{\mathrm {PS}_3})$ and so it is not a maximal paraconsistent logic with respect to $\mathrm {CPL}$ . But still we have the following theorem.

The logic of the algebra-valued model $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {H}_3)}$ is not $\mathrm {CPL}$ : for $\varphi \in \mathcal {L}_\Lambda $ , the formula $\varphi \vee \lnot \varphi \notin \mathbf {L}(\mathbb {H}_3, D_{\mathbb {H}_3}) = \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathbb {H}_3}, D_{\mathbb {H}_3})$ , which implies $\varphi \vee \lnot \varphi \notin \mathbf {L}([\kern-1.5pt[ \cdot ]\kern-1.5pt] _{\mathrm {PS}_3 \times \mathbb {H}_3}, D_{\mathrm {PS}_3 \times \mathbb {H}_3})$ , by Theorem 4.28, but $\varphi \vee \lnot \varphi \in \mathrm {CPL}$ .

We can derive from Theorem 4.29 that $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {H}_3)}$ is loyal to $(\mathrm {PS}_3 \times \mathbb {H}_3, D_{\mathrm {PS}_3 \times \mathbb {H}_3})$ , as $\mathbf {V}^{(\mathrm {PS}_3)}$ and $\mathbf {V}^{(\mathbb {H}_3)}$ are loyal to $(\mathrm {PS}_3, D_{\mathrm {PS}_3})$ and $(\mathbb {H}_3, D_{\mathbb {H}_3})$ , respectively. We will further show that $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {H}_3)}$ is faithful to the algebra $\mathrm {PS}_3 \times \mathbb {H}_3$ as well.

It is not hard to see that if one of the component algebras $\mathbb {A}$ (say) is such that $\mathbf {V}^{(\mathbb {A})}$ is not faithful to $\mathbb {A}$ then the product-algebra-valued model will also not be faithful to the product algebra. But, the following question is still open.

We end Section 4 with Table 2, which displays in one look the loyalty and faithfulness of a product-algebra-valued models.

Table 2 Loyalty and faithfulness of product algebra-valued model

5.1 Sentences which inherit independence from classical set theory

We shall first prove one of the main results of the paper in its most general form and then we will apply it in the context of $\mathbb {L}\mathrm {PS}_3$ .

Proof. Let $\mathbb {B}_1, \mathbb {B}_2$ be two Boolean algebras such that $\mathbf {V}^{(\mathbb {B}_1)} \models \varphi $ but $\mathbf {V}^{(\mathbb {B}_2)} \not \models \varphi $ . We know that $D_{\mathbb {A} \times \mathbb {B}_1} = D_{\mathbb {A}} \times D_{\mathbb {B}_1}$ be the designated set of $\mathbb {A} \times \mathbb {B}_1$ , where $D_{\mathbb {A}}$ and $D_{\mathbb {B}_1}$ are the designated sets of $\mathbb {A}$ and $\mathbb {B}_1$ , respectively. By our assumption,

$$\begin{align*}[\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{A} \times \mathbb{B}_1} = ([\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{A}}, [\kern-1.5pt[ \varphi ]\kern-1.5pt]_{\mathbb{B}_1}) \in D,\end{align*}$$

i.e., $\mathbf {V}^{(\mathbb {A} \times \mathbb {B}_1)} \models \varphi $ . On the other hand, since $[\kern-1.5pt[ \varphi ]\kern-1.5pt] _{\mathbb {B}_2} \notin D_{\mathbb {B}_2}$ , where $D_{\mathbb {B}_2}$ is the designated set of $\mathbb {B}_2$ , we can conclude that $\mathbf {V}^{( \mathbb {A} \times \mathbb {B}_2)} \not \models \varphi $ . By our assumption we also get that $\mathbf {L}(\mathbb {A} \times \mathbb {B}_i, D_{\mathbb {A} \times \mathbb {B}_1}) = \mathcal {L}_{\Lambda } \cap \mathrm {CPL} = \mathcal {L}_{\Lambda },$ for $i = 1, 2$ . Moreover, by Observation 3.22 we get that $\mathrm {ValSent}_{\mathbb {A} \times \mathbb {B}_i} = \mathrm {ValSent}_{\mathbb {A}} \cap \mathrm {ValSent}_{\mathbb {B}_i}$ , for $i = 1, 2$ . Hence, by our assumption, $\mathsf {T} \subseteq \mathrm {ValSent}_{\mathbb {A} \times \mathbb {B}_i} \subsetneqq \mathsf {BZF}$ , i.e., $\mathbf {V}^{(\mathbb {A} \times \mathbb {B}_i)} \models \mathsf {T}$ but $\mathbf {V}^{(\mathbb {A} \times \mathbb {B}_i)} \not \models \mathsf {BZF}$ , for $i = 1, 2$ .□

Notice that the result above shows not only the formal independence of $\varphi $ with respect to a proper fragment $\mathsf {T}$ of $\mathsf {BZF}$ , but also that the independence of $\varphi $ is carried out in models that validate only the weaker theory $\mathsf {T}$ . Therefore, Theorem 5.35 is telling us more than the trivial observation that independence is preserved in weaker theories.

To get a more concrete sense of this observation, let us consider a theory $\mathsf {T}$ which is a proper fragment of both $\mathsf {BZF}$ and $\mathsf {ZF}$ . In other words, there are axioms of $\mathsf {ZF}$ that are not contained in (the deductive closure of) $\mathsf {T}$ . To simplify the exposition, let us suppose that this axiomatic difference consists of an axiom $\theta $ and consider an algebra $\mathbb {A}$ such that the algebra-valued model $\mathbf {V}^{(\mathbb {A})}$ validates $\mathsf {T}$ and $\varphi $ , but not $\theta $ , for a sentence $\varphi $ as in the proof of Theorem 5.35: i.e., provably independent from $\mathsf {BZF}$ using two Boolean algebras $\mathbb {B}_1, \mathbb {B}_2$ . Then we get the following:

1. (i) $\mathbf {V}^{(\mathbb {A} \times \mathbb {B}_1)} \models \mathsf {T}$ and $\mathbf {V}^{(\mathbb {A} \times \mathbb {B}_2)} \models \mathsf {T}$ ,

2. (ii) $\mathbf {V}^{(\mathbb {A} \times \mathbb {B}_1)} \models \varphi $ and $\mathbf {V}^{(\mathbb {A} \times \mathbb {B}_2)} \not \models \varphi $ ,

3. (iii) $\mathbf {V}^{(\mathbb {A} \times \mathbb {B}_1)} \not \models \theta $ and $\mathbf {V}^{(\mathbb {A} \times \mathbb {B}_2)} \not \models \theta $ .

In this way it is possible to show that the independence of $\varphi $ with respect to $\mathsf {T}$ does not need the axiom $\theta $ (and thus the full strength of $\mathsf {ZF}$ ). In this sense, Theorem 5.35 can provide a sort of reverse analysis for independence proofs, providing models of weaker theories for independence results.

There is an even stronger version of this phenomenon that is captured by the following definition.

Notice that Definition 5.36 can capture a phenomenon that can be recast in syntactic terms. Indeed, if $\theta $ is superfluous for the independence of $\varphi $ from $\mathsf {T}$ , then $\varphi $ is actually independent from the theory $\mathsf {T} \cup \{\neg \theta \}$ . This observation becomes relevant once $\theta $ is taken to be an axiom of $\mathsf {ZF}$ . Indeed, in this case, $\mathsf {T} \cup \{\neg \theta \}$ is not anymore a fragment of $\mathsf {ZF}$ . Therefore, a proof of independence from such a theory represents a result that cannot be obtained using Boolean-valued models.

We will now provide a concrete application of algebra-valued models in the proof of the independence of $\mathsf {CH}$ in the context of the logic $\mathbb {L}\mathrm {PS}_3$ . We will provide two such proofs. The first uses Theorem 5.35, while the second uses the specific set-theoretical properties of $\mathbf {V}^{(\mathrm {PS}_3)}$ . The reason for a second proof is to be found in the possibility to provide a concrete example of the phenomenon captured by Definition 5.36

5.2 The independence of $\mathsf {CH}$

In this section we offer an important example of independence in non-classical set theory: the independence of $\mathsf {CH}$ from a set theory whose underlying logic is $\mathbb {L}\mathrm {PS}_3$ . We will provide two proofs of this fact. One using Theorem 5.35 and one using a detailed study of cardinality results in $\mathbf {V}^{(\mathrm {PS}_3)}$ . Toward this end we first need to properly define the set theory from which $\mathsf {CH}$ will be proved independent.

In this section, we will work with $\mathrm {PS}_3\mbox {-}\mathsf {ZF}$ .

Proof. (i) By Theorem 2.7, we know that $\mathbf {V}^{(\mathrm {PS}_3)} \models \mathrm {NFF}$ - $\mathsf {ZF}$ . It is also clear that for any complete Boolean algebra $\mathbb {B}$ , $\mathbf {V}^{(\mathbb {B})} \models \mathrm {NFF}$ - $\mathsf {ZF}$ . Hence, we can conclude that $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {B})} \models \mathrm {NFF}$ - $\mathsf {ZF}$ , where $\mathbb {B}$ is any complete Boolean algebra.

(ii) Immediate from the definitions of $\mathrm {PS}_3\mbox {-}\mathsf {ZF}$ and $\mathsf {BZF}$ .

(iii) In Section §3.2, we have already discussed that , which implies that as well. Now, consider the formula $\varphi \in \mathrm {Sent}_{\Lambda , \in }$ as follows:

$$\begin{align*}\varphi: = (\mathsf{Par} \wedge \neg \mathsf{Par}) \to \bot.\end{align*}$$

Clearly, . Hence, independent of the choice of a complete Boolean algebra $\mathbb {B}$ , it can be concluded that $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {B})} \not \models \varphi $ , i.e., $\varphi \notin \mathrm {PS}_3\mbox {-}\mathsf {ZF}$ . Thus, by the definition of paraconsistency, we get that $\mathrm {PS}_3\mbox {-}\mathsf {ZF}$ is a paraconsistent set theory.□

The general structure of the proof of the independence of $\mathsf {CH}$ from $\mathrm {PS}_3\mbox {-}\mathsf {ZF}$ can be summarized as follows. There are two $\Lambda $ -algebras $\mathbb {A}$ , $\mathbb {B}$ , such that the following hold:

1. (i) $\mathbf {L}(\mathbb {A}, D_{\mathbb {A}}) = \mathbf {L}(\mathbb {B}, D_{\mathbb {B}}) = \mathbb {L}\mathrm {PS}_3$ ,

2. (ii) $\mathbf {V}^{(\mathbb {A})} \models \mathrm {PS}_3\mbox {-}\mathsf {ZF}$ , $\mathbf {V}^{(\mathbb {B})} \models \mathrm {PS}_3\mbox {-}\mathsf {ZF}$ ,

3. (iii) $\mathbf {V}^{(\mathbb {A})} \models \varphi $ but $\mathbf {V}^{(\mathbb {B})} \not \models \varphi $ .

Let us now develop some cardinal notions in $\mathbf {V}^{(\mathrm {PS}_3)}$ in order to show that $\mathsf {CH}$ is still a meaningful sentence in $\mathrm {PS}_3\mbox {-}\mathsf {ZF}$ , which really expresses what we expect it to express.

5.2.1 Cardinality in $\mathbf {V}^{(\mathrm {PS}_3)}$ and the Continuum Hypothesis

For notational simplicity, let D be the designated set of $\mathrm {PS}_3$ . For any $u \in \mathbf {V}^{(\mathrm {PS}_3)}$ , the subset $\mathrm {dom}_D(u)$ of $\mathrm {dom}(u)$ is defined as follows: $x \in \mathrm {dom}_D(u)$ iff $u(x) \in D$ . It was shown in [Reference Löwe and Tarafder13] that the relation $\sim $ , defined as $u \sim v$ iff $\mathbf {V}^{(\mathrm {PS}_3)} \models u = v$ , is an equivalence relation. Hence, $\mathrm {dom}_D(u)$ can be partitioned by $\sim $ into equivalent classes. Let $\mathrm {Part}(\mathrm {dom}_D(u)) = \mathrm {dom}_D(u)/\sim $ , the quotient (or partition) of $\mathrm {dom}_D(u)$ by $\sim $ . If $\mathrm {InjFunc}(f; x, y), \mathrm {SurjFunc}(f; x, y)$ , and $\mathrm {BijFunc}(f; x, y)$ stand for the first order formulas stating that f is an injection, surjection, and bijection from x into y, respectively, then the following theorem holds.

Theorem 5.39 [Reference Tarafder17].

If $\mathbf {V}$ is a model of $\mathsf {ZFC}$ , then for any two elements $u, v \in \mathbf {V}^{(\mathrm {PS}_3)}$ ,

1. (i) there exists an injection between $\mathrm {Part}(\mathrm {dom}_D(u))$ and $\mathrm {Part}(\mathrm {dom}_D(v))$ in $\mathbf {V}$ if and only if $\mathbf {V}^{(\mathrm {PS}_3)} \models \exists f \mathrm {InjFunc}(f; x, y)$ ,

2. (ii) there exists a surjection between $\mathrm {Part}(\mathrm {dom}_D(u))$ and $\mathrm {Part}(\mathrm {dom}_D(v))$ in $\mathbf {V}$ if and only if $\mathbf {V}^{(\mathrm {PS}_3)} \models \exists f \mathrm {SurjFunc}(f; x, y)$ ,

3. (iii) there exists a bijection between $\mathrm {Part}(\mathrm {dom}_D(u))$ and $\mathrm {Part}(\mathrm {dom}_D(v))$ in $\mathbf {V}$ if and only if $\mathbf {V}^{(\mathrm {PS}_3)} \models \exists f \mathrm {BijFunc}(f; x, y)$ .

Let us denote $\mathrm {ORD}$ as the class of all ordinal numbers in $\mathbf V$ . Then for each $\alpha \in \mathrm {ORD}$ the $\alpha $ -like elements in $\mathbf {V}^{(\mathrm {PS}_3)}$ are defined by transfinite recursion as follows.

Definition 5.40 [Reference Tarafder, Banerjee and Krishna16].

An element $x \in \mathbf {V}^{(\mathrm {PS}_3)}$ is called:

1. (i) $0$ -like if $x(y) = \mathbf {0}$ for any $y \in \mathrm {dom}(x)$ ,

2. (ii) $\alpha $ -like for some $\alpha \in \mathrm {ORD}$ if for each $\beta \in \alpha $ there exists $y \in \mathrm {dom}(x)$ which is $\beta $ -like and , and for any $z \in \mathrm {dom}(x)$ if it is not $\beta $ -like for any $\beta \in \alpha $ then $x(z) = \mathbf {0}$ ,

3. (iii) ordinal-like if it is $\alpha $ -like, for some $\alpha \in \mathrm {ORD}$ .

Let $\mathsf {Ord}(x)$ be the first order formula in $\mathcal {L}_{\Lambda , \in }$ , naively states that x is an ordinal number. Then we have the following theorem.

Theorem 5.41 [Reference Tarafder, Banerjee and Krishna16, theorem 13].

Let $\alpha \in \mathrm {ORD}$ and u be an $\alpha $ -like element in $\mathbf {V}^{(\mathrm {PS}_3)}$ . Then $\mathbf {V}^{(\mathrm {PS}_3)} \models \mathsf {Ord}(u)$ .

Theorem 5.42 [Reference Tarafder17].

Let $u \in \mathbf {V}^{(\mathrm {PS}_3)}$ be any element and $\kappa $ be the cardinality of $\mathrm {Part}(\mathrm {dom}_D(u))$ in $\mathbf {V}$ , a model of $\mathsf {ZFC}$ . Then in $\mathbf {V}^{(\mathrm {PS}_3)}$ , there exist bijections between u and $\kappa $ -like elements, but there does not exist any bijection between u and any $\alpha $ -like element, where $\alpha < \kappa $ in $\mathbf {V}$ .

Proof. Follows from Theorem 5.39.□

Let $\mathsf {Card}(x)$ be the first order formula expressing that ‘x is a cardinal number’. Then, for a given $u \in \mathbf {V}^{(\mathrm {PS}_3)}$ , consider the following collection of names of the cardinal number of u in $\mathbf {V}^{(\mathrm {PS}_3)}$ :

$$\begin{align*}\mathsf{Cardinal}_u := \{v \in \mathbf{V}^{(\mathrm{PS}_3)}: \mathbf{V}^{(\mathrm{PS}_3)} \models \mathsf{Card}(v) \wedge \exists f \mathrm{BijFunc}(f;u, v)\}.\end{align*}$$

Any element of $\mathsf {Cardinal}_u$ will be called a name of the cardinal number of u or simply a cardinal number of u in $\mathbf {V}^{(\mathrm {PS}_3)}$ .

For each $x \in \mathbf V$ , let us define an element $\hat {x}$ of $\mathbf {V}^{(\mathbb {A})}$ recursively as: $\hat {\varnothing } = \varnothing $ and $\hat {x} = \{\langle \hat {y}, \mathbf {1} \rangle : y \in x\}$ . It is proved in [Reference Tarafder17] that, if $u \in \mathbf {V}^{(\mathrm {PS}_3)}$ is an element such that the cardinality of $\mathrm {Part}(\mathrm {dom}_D(u))$ in $\mathbf {V}$ is $\kappa $ , then $\mathsf {Cardinal}_u$ is the class of all $\kappa $ -like elements and hence, in particular, $\hat {\kappa } \in \mathsf {Cardinal}_u$ . Moreover, for any two elements $u,v \in \mathbf {V}^{(\mathrm {PS}_3)}$ , $\mathbf {V}^{(\mathrm {PS}_3)} \models \exists f \mathrm {BijFunc}(f;u, v)$ if and only if $\mathsf {Cardinal}_u = \mathsf {Cardinal}_v$ .

Definition 5.43 [Reference Tarafder17].

For two elements $u, v \in \mathbf {V}^{(\mathrm {PS}_3)}$ , it will be said that the cardinality of u is less than the cardinality of v, denoted by $|u|_{\mathbf {V}^{(\mathrm {PS}_3)}} < |v|_{\mathbf {V}^{(\mathrm {PS}_3)}}$ , if for any $p \in \mathsf {Cardinal}_u$ and $q \in \mathsf {Cardinal}_v$ , $\mathbf {V}^{(\mathrm {PS}_3)} \models p \in q$ .

In [Reference Tarafder17], it is proved that for any pair $u, v \in \mathbf {V}^{(\mathrm {PS}_3)}$ , not only the notion of $|u|_{\mathbf {V}^{(\mathrm {PS}_3)}} < |v|_{\mathbf {V}^{(\mathrm {PS}_3)}}$ is well-defined, but also that $|u|_{\mathbf {V}^{(\mathrm {PS}_3)}} < |v|_{\mathbf {V}^{(\mathrm {PS}_3)}}$ holds if and only if $\mathbf {V}^{(\mathrm {PS}_3)} \models \exists f \mathrm {InjFunc}(f;u, v) \mbox { but } \mathbf {V}^{(\mathrm {PS}_3)} \not \models \exists f \mathrm {InjFunc}(f;v, u)$ .

Theorem 5.44 [Reference Tarafder17].

For a model $\mathbf {V}$ of $\mathsf {ZFC}$ , $\mathbf {V} \models \mathsf {CH}$ if and only if $\mathbf {V}^{(\mathrm {PS}_3)} \models \mathsf {CH}$ .

Proof. Let $\mathbf {V} \models \mathsf {CH}$ . Suppose $u \in \mathbf {V}^{(\mathbb {A})}$ be an $\omega $ -like element and $v \in \mathbf {V}^{(\mathrm {PS}_3)}$ be a name for the power set of u in $\mathbf {V}^{(\mathrm {PS}_3)}$ . Then, it can be proved that $\hat {\aleph }_0 \in \mathsf {Cardinal}_u$ and $\hat {(2^{\aleph _0})} \in \mathsf {Cardinal}_v$ . If there exist $s \in \mathbf {V}^{(\mathrm {PS}_3)}$ such that $|u|_{\mathbf {V}^{(\mathrm {PS}_3)}} < |s|_{\mathbf {V}^{(\mathrm {PS}_3)}} < |v|_{\mathbf {V}^{(\mathrm {PS}_3)}}$ and $\hat {\kappa } \in \mathsf {Cardinal}_s$ , where $\kappa $ is a cardinal number in $\mathbf {V}$ , then we can conclude that $\hat {\aleph }_0 \in \hat {\kappa } \in \hat {(2^{\aleph _0})}$ holds in $\mathbf {V}^{(\mathrm {PS}_3)}$ . Hence, $\aleph _0 < \kappa < 2^{\aleph _0}$ holds in $\mathbf {V}$ , which contradicts our assumption $\mathbf {V} \models \mathsf {CH}$ .

Conversely, suppose $\mathbf {V} \not \models \mathsf {CH}$ . Then there exists a cardinal number $\kappa $ in $\mathbf {V}$ such that $\aleph _0 < \kappa < 2^{\aleph _0}$ . Hence, $\hat {\aleph }_0 \in \hat {\kappa } \in \hat {(2^{\aleph _0})}$ holds in $\mathbf {V}^{(\mathrm {PS}_3)}$ , which implies that $\mathbf {V}^{(\mathrm {PS}_3)} \not \models \mathsf {CH}$ .□

We are now in the position to state the independence result for $\mathsf {CH}$ .

Theorem 5.45. There are two algebra-valued models of $\mathrm {PS}_3\mbox {-}\mathsf {ZF}$ , and not of $\mathsf {BZF}$ , which do not agree on the validity of $\mathsf {CH}$ , thus showing the independence of $\mathsf {CH}$ from $\mathrm {PS}_3\mbox {-}\mathsf {ZF}$ .

Proof. Let $\mathbb {B}_1, \mathbb {B}_2$ be two Boolean algebras such that $\mathbf {V}^{(\mathbb {B}_1)} \models \mathsf {CH}$ and $\mathbf {V}^{(\mathbb {B}_2)} \not \models \mathsf {CH}$ and, without loss of generality, let us assume that $\mathbf {V} \models \mathsf {CH}$ (if not, i.e., $\mathbf {V} \models \neg \mathsf {CH}$ , we just need to switch the role of $\mathbb {B}_1$ and $\mathbb {B}_2$ in the rest of the proof). Then using Theorems 5.35 and 5.44 we can conclude that:

1. (i) $\mathbf {L}(\mathrm {PS}_3 \times \mathbb {B}_1, D_{\mathrm {PS}_3 \times \mathbb {B}_1}) = \mathbf {L}(\mathrm {PS}_3 \times \mathbb {B}_2, D_{\mathrm {PS}_3 \times \mathbb {B}_2}) = \mathbb {L}\mathrm {PS}_3$ ,

2. (ii) $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {B}_1)} \models \mathrm {PS}_3\mbox {-}\mathsf {ZF}$ , $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {B}_2)} \models \mathrm {PS}_3\mbox {-}\mathsf {ZF}$ , and

3. (iii) $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {B}_1)} \models \mathsf {CH}$ , but $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {B}_2)} \not \models \mathsf {CH}$ .

Then, remember that $\mathsf {Sep}$ is the instance of the $\mathsf {Separation\ Axiom}$ schema that, as shown in Theorem 2.8, fails in $\mathbf {V}^{(\mathrm {PS}_3)}$ . Because of the coordinate-wise definition of validity in product algebras, we get that $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {B}_1)}\not \models \mathsf {Sep}$ and $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {B}_2)}\not \models \mathsf {Sep}$ . Hence $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {B}_1)}$ and $\mathbf {V}^{(\mathrm {PS}_3 \times \mathbb {B}_2)}$ are two algebra-valued models of $\mathrm {PS}_3\mbox {-}\mathsf {ZF}$ , but not of $\mathsf {BZF}$ , witnessing the independence of $\mathsf {CH}$ .□

It is possible to improve Theorem 5.45 by providing a more direct proof of the independence of $\mathsf {CH}$ that uses only the ability to transfer cardinal properties from $\mathbf {V}$ to $\mathbf {V}^{(\mathrm {PS}_3)}$ .

Theorem 5.46. $\mathsf {CH}$ is independent from $\mathrm {NFF}$ - $\mathsf {ZF}\cup \{\neg \mathsf {Sep}\}$ .

Proof. It is sufficient to consider two classical models of $\mathsf {ZFC}$ , $\mathbf {V}_1$ and $\mathbf {V}_2$ such that $\mathbf {V}_1 \models \mathsf {CH}$ and $\mathbf {V}_2 \models \neg \mathsf {CH}$ and then to build an algebra-valued model with values in $\mathrm {PS}_3$ over each of these two classical structures, say $\mathbf {V}_1^{(\mathrm {PS}_3)}$ and $\mathbf {V}_2^{(\mathrm {PS}_3)}$ . Then, because the validity of Theorem 2.7 only depends on the fact that the ground model satisfies $\mathsf {ZFC}$ , we get that both $\mathbf {V}_1^{(\mathrm {PS}_3)}$ and $\mathbf {V}_2^{(\mathrm {PS}_3)}$ validate $\mathrm {NFF}$ - $\mathsf {ZF}$ . Moreover, Theorem 2.8 can be applied in both $\mathbf {V}_1^{(\mathrm {PS}_3)}$ and $\mathbf {V}_2^{(\mathrm {PS}_3)}$ to provide the validity of $\neg \mathsf {Sep}$ . Finally, Theorem 5.44 yields $\mathbf {V}_1^{(\mathrm {PS}_3)} \models \mathsf {CH}$ and $\mathbf {V}_2^{(\mathrm {PS}_3)} \not \models \mathsf {CH}$ .□

Notice that Theorem 5.46 states the independence of $\mathsf {CH}$ with respect to $\mathrm {NFF}$ - $\mathsf {ZF}\cup \{\neg \mathsf {Sep}\}$ and not $\mathrm {PS}_3\mbox {-}\mathsf {ZF}\cup \{\neg \mathsf {Sep}\}$ . The reason is that the theory of an algebra-valued model based on $\mathrm {PS}_3$ depends on the theory of the ground model (as shown in Theorem 5.44). Our definition of $\mathrm {PS}_3\mbox {-}\mathsf {ZF}$ is given in terms of (the real) $\mathbf {V}$ as the ground model, but nothing forbids us to define a similar theory in terms of a different ground model. It is unknown to the authors if the definition of $\mathrm {PS}_3\mbox {-}\mathsf {ZF}$ is invariant on the choice of a ground model.Footnote ⁸ For this reason we decided to state Theorem 5.46 in terms of $\mathrm {NFF}$ - $\mathsf {ZF}$ . However, we should remark that the proof of Theorem 5.46 yields the independence of $\mathsf {CH}$ from the theory that is stronger than $\mathrm {NFF}$ - $\mathsf {ZF}\cup \{\neg \mathsf {Sep}\}$ , namely, $Th_\in (\mathbf {V}_1^{(\mathrm {PS}_3)}) \cap Th_\in (\mathbf {V}_2^{(\mathrm {PS}_3)})$ , where $Th_\in (\mathcal {M})$ is the theory (in the model theoretic sense) of $\mathcal {M}$ , in the signature of the pure language of set theory i.e., of $\mathcal {L}_{\Lambda , \in }$ .

Now, considering that $\mathsf {CH}$ is independent from $\mathsf {BZF}$ , using the standard Cohen construction, we get the following corollary.

Corollary 5.47. There are instances of $\mathsf {Separation\ Axiom}$ that are superfluous for the proof of independence of $\mathsf {CH}$ from $\mathrm {NFF}$ - $\mathsf {ZF}$ .

The example of $\mathsf {CH}$ has given an example of the preservation of independence from the classical to the non-classical case. We now turn to the study of set theoretic sentences which are only independent with respect to proper fragments of $\mathsf {BZF}$ .

5.3 Sentences independent from non-classical set theory only

Let $\mathbb {A}$ be a $\Lambda $ -algebra and recall that, by definition, $\mathbb {A}$ - $\mathsf {ZF}$ is a first order fragment of $\mathsf {BZF}$ . The second main result of the paper states the following.

Notice that there is a small difference between the two possibilities described in Theorem 5.48 Indeed, while in the proof of (i) both models witnessing the independence of $\varphi $ are only models of the smaller theory $\mathbb {A}$ - $\mathsf {ZF}$ , this is not the case anymore in (ii). As a matter of fact, in (ii) we need to resort to a classical Boolean-valued model, which being a model of $\mathsf {BZF}$ is a fortiori a model of $\mathbb {A}$ - $\mathsf {ZF}$ . However, it is important to stress the importance of (ii) from a more conceptual perspective, since it shows that the strength of a stronger theory is only a matter of solving independence from the perspective of a weaker theory.