Constructing illoyal algebra-valued models of set theory

BENEDIKT LOWE

Institute for Logic, Language and Computation, Universiteit van AmsterdamFachbereich Mathematik, Universitat Hamburg

Churchill College, University of Cambridge

and

ROBERT PASSMANN

Institute for Logic, Language and Computation, Universiteit van Amsterdam

and

SOURAV TARAFDER

Department of Mathematics, St. Xavier’s College, Kolkata

Abstract. An algebra-valued model of set theory is called loyal to its algebra if themodel and its algebra have the same propositional logic; it is called faithful if all elementsof the algebra are truth values of a sentence of the language of set theory in the model.We observe that non-trivial automorphisms of the algebra result in models that are notfaithful and apply this to construct three classes of illoyal models: the tail stretches, thetransposition twists, and the maximal twists. (Version 7; 2 October 2018)

§1. Background The construction of algebra-valued models of set theory startsfrom an algebra A and a model V of set theory and forms an A-valued model of settheory that reflects both the set theory of V and the logic of A. This constructionis the natural generalisation of Boolean-valued models, Heyting-valued models,lattice-valued models, and orthomodular-valued models (Bell, 2005; Grayson, 1979;Ozawa, 2009; Titani, 1999) and was developed by Lowe and Tarafder (2015).

Lowe and Tarafder (2015, § 6) used an algebra PS3 of paraconsistent logic to con-struct a PS3-valued model of set theory that exhibits the paraconsistency inheritedfrom the algebra PS3. For more on the algebra PS3, cf. (Chakraborty and Tarafder,2016); for more on the set theory in the PS3-valued model, cf. (Tarafder, 2015).

Recently, Passmann (2018) introduced the terms “loyalty” and “faithfulness”while studying the precise relationship between the logic of the algebra A and thelogical phenomena witnessed in the A-valued model of set theory. A model is calledloyal to its algebra if the propositional logic in the model is the same as the logicof the algebra from which it was constructed and faithful if every element of thealgebra is the truth value of a sentence in the model. The model constructed byLowe and Tarafder (2015) is both loyal and faithful to PS3. In this paper, we shallgive elementary constructions to produce illoyal models by stretching and twistingBoolean algebras.

This research was partially supported by the Marie Sk lodowska-Curie fellowshipREGPROP (706219) funded by the European Commission at the Universitat Hamburg.The authors would like to thank Nick Bezhanishvili and Lorenzo Galeotti for variousdiscussions about Heyting algebras and their logics.

2 B. Lowe, R. Passmann, S. Tarafder

After we give the basic definitions in §2., we remind the reader of the construc-tion of algebra-valued models of set theory in §3.. In §4., we introduce our maintechnique: a non-trivial automorphisms of an algebra A excludes values from beingtruth values of sentences in the A-valued model of set theory (Corollary 7). Finally,in §5., we apply this technique to produce three classes of models: tail stretches(§ 5.2.), transposition twists (§ 5.3.), and maximal twists (§ 5.4.).

§2. Basic definitions

Algebras. Let Λ be a set of logical connectives; in this paper, we shall assumethat

{∧,∨,0,1} ⊆ Λ ⊆ {∧,∨,→,¬,0,1}.An algebra A with underlying set A is called a Λ-algebra if it has one operationfor each of the logical connectives in Λ such that (A,∧,∨,0,1) is a distributivelattice;1 we can define ≤ on A by x ≤ y if and only if x ∧ y = x. An element a ∈ Ais an atom if it is ≤-minimal in A\{0}; we write At(A) for the set of atoms in A. IfΛ = {∧,∨,→,0,1}, we call A an implication algebra and if Λ = {∧,∨,→ ¬,0,1},we call A an implication-negation algebra.

We call a Λ-algebra A with underlying set A complete if for every X ⊆ A, the≤-supremum and ≤-infimum exist; in this case, we write

∨X and

∧X for these

elements of A. A complete Λ-algebra A is called atomic if for every a ∈ A, there isan X ⊆ At(A) such that a =

∨X.

Boolean algebras, complementation, & Heyting algebras. An algebra B =(B,∧,∨,¬,0,1) is called a Boolean algebra if for all b ∈ B, we have that b∧¬b = 0and b ∨ ¬b = 1. As usual, we can define an implication by

x→ y := ¬x ∨ y; (#)

using this definition, we can consider Boolean algebras as implication algebras orimplication-negation algebras. An implication algebra (B,∧,∨,→,0,1) is called aBoolean implication algebra if there is a Boolean algebra (B,∧,∨,¬,0,1) such that→ is defined by (#) from ∨ and ¬ or, equivalently, if the negation defined by¬∗x := x→ 0 satisfies ¬∗b ∧ b = 0 and ¬∗b ∨ b = 1.

On an atomic distributive lattice A = (A,∧,∨,0,1), we have a canonical defi-nition for a negation operation, the complementation negation: since A is atomic,every element a ∈ A is uniquely represented by a set X ⊆ At(A) such that a =

∨X.

Then we define the complementation negation by

¬c(∨X) :=

∨{t ∈ At(A) ; t /∈ X}.

In this situation, (A,∧,∨,¬c,0,1) is an atomic Boolean algebra. Moreover, if(A,∧,∨,¬,0,1) is an atomic Boolean algebra and ¬c is the complementation nega-tion of the atomic distributive lattice (A,∧,∨,0,1), then ¬ = ¬c. Of course, for

1 As usual, we use the same notation for the syntactic logical connectives and theoperations on A interpreting them. In the rare cases where proper marking of thesesymbols improves readability, we attach a subscript A to the algebra operations in A,e.g., ∧A, ∨A,

∧A, or

∨A.

Constructing illoyal models of set theory 3

every set X, the power set algebra (℘(X),∩,∪,∅, X) forms an atomic distributivelattice and, with the set complementation operator, a Boolean algebra.

If (H,∧,∨,0,1) is a complete distributive lattice, then an implication algebraH = (H,∧,∨,→,0,1) is called a complete Heyting algebra if and only if for alla, b ∈ H, we have that

a→ b =∨{x ∈ H ; a ∧ x ≤ b}.

In a Heyting algebra H, we can define a negation ¬H by ¬Hx := x→ 0. Note thatBoolean implication algebras are Heyting algebras.

A Heyting algebra is called linear if (H,≤) is a linear order; Horn (1969) showedthat the formula (p → q) ∨ (q → p) characterises the variety of Heyting algebrasgenerated by the linear Heyting algebras.

We shall later use the following linear three element Heyting algebra3 := ({0, 1/2,1},∧,∨,→,0,1) with 0 ≤ 1/2 ≤ 1 and → defined by

→ 0 1/2 10 1 1 1

1/2 0 1 11 0 1/2 1 .

Languages. Fix a set S of non-logical symbols, a countable set P of propositionalvariables, and a countable set V of first-order variables. We denote the propositionallogic with connectives Λ and propositional variable P by LΛ and the first-order logicwith connectives Λ, variables in V and constant, relation and function symbols inS by LΛ,S . The subset of sentences of LΛ,S will be denoted by SentΛ,S . Note thatboth LΛ and SentΛ,S have the structure of a Λ-algebra and that the Λ-algebra LΛ

is generated by closure under the connectives in Λ from the set P .For arbitrary sets Λ of logical connectives and S of non-logical symbols, we define

NFFΛ to be the closure of P under the logical connectives other than ¬ and NFFΛ,S

to be the closure of the atomic formulae of LΛ,S under the logical connectives otherthan ¬. These formulas are called the negation-free Λ-formulas. Clearly, if ¬ /∈ Λ,then LΛ = NFFΛ and LΛ,S = NFFΛ,S .

Homomorphisms, assignments, & translations. For any two Λ-algebras Aand B, a map f : A→ B is called a Λ-homomorphism if it preserves all connectivesin Λ; it is called a Λ-isomorphism if it is a bijective Λ-homomorphism; isomorphismsfrom A to A are called Λ-automorphisms.

If A and B are two complete Λ-algebras and f : A → B is a Λ-homomorphism,we call it complete if it preserves the operations

∨and

∧, i.e., f(

∨A(X)) =∨

B({f(x) ; x ∈ X}) and f(∧

A(X)) =∧

B({f(x) ; x ∈ X}) for X ⊆ A.Since LΛ is generated from P , we can think of any Λ-homomorphism defined

on LΛ as a function on P , homomorphically extended to all of LΛ. If A is a Λ-algebra with underlying set A, we say that Λ-homomorphisms ι : LΛ → A areA-assignments; if S is a set of non-logical symbols, we say that Λ-homomorphismsT : LΛ → SentΛ,S are S-translations.

The propositional logic of an algebra. A set D ⊆ A is called a designated setif the following three conditions hold: (i) 1 ∈ D, (ii) 0 /∈ D, and (iii) if x ∈ D andx ≤ y, then y ∈ D. A designated set D is called a filter if, in addition, (iv) for

4 B. Lowe, R. Passmann, S. Tarafder

x, y ∈ D, we have x ∧ y ∈ D. For any designated set D, the propositional logic of(A, D) is defined as

L(A, D) := {ϕ ∈ LΛ ; ι(ϕ) ∈ D for all A-assignments ι}.

Note that if B is a Boolean algebra and D is any filter, then L(B, D) = CPC, theclassical propositional calculus (Blackburn et al., 2001, Theorem 5.11).

Algebra-valued structures and their propositional logic. If A is a Λ-algebraand S is a set of non-logical symbols, then any Λ-homomorphism J·K : SentΛ,S → Awill be called an A-valued S-structure. Note that if S′ ⊆ S and J·K is an A-valuedS-structure, then J·K�SentΛ,S′ is an A-valued S′-structure.

We define the propositional logic of (J·K, D) as

L(J·K, D) := {ϕ ∈ LΛ ; JT (ϕ)K ∈ D for all S-translations T}.

Note that if T is an S-translation and J·K is an A-valued S-structure, then ϕ 7→JT (ϕ)K is an A-assignment, so

L(A, D) ⊆ L(J·K, D). (†)

Clearly, ran(J·K) ⊆ A is closed under all operations in Λ (since J·K is a homomor-phism) and thus defines a sub-Λ-algebra AJ·K of A. The A-assignments that are ofthe form ϕ 7→ JT (ϕ)K are exactly the AJ·K-assignments, so we obtain

L(J·K, D) = L(AJ·K, D).

Loyalty & faithfulness. An A-valued S-structure J·K is called loyal to (A, D) ifthe converse of (†) holds, i.e., L(A, D) = L(J·K, D); it is called faithful to A if forevery a ∈ A, there is a ϕ ∈ SentΛ,S such that JϕK = a; equivalently, if AJ·K = A. Thetwo notions central for our paper were introduced by Passmann (2018) in a moregeneral setting for classes of so-called Heyting structures in the sense of Fourmanand Scott (1979) (cf. Passmann, 2018, Definitions 2.39 & 2.40). In this paper, weshall not need the level of generality provided in (Passmann, 2018) and stick to theabove simplified definitions.

Lemma 1 Let Λ be any set of propositional connectives, S be any set of non-logicalsymbols, A be a Λ-algebra, and J·K be an A-valued S-structure. Then, if J·K is faithfulto A, then it is loyal to (A, D) for any designated set D.

Proof. By (†), we only need to prove one inclusion; if ϕ /∈ L(A, D), then letp1, . . . , pn be the propositional variables occurring in ϕ and let ι be an assignmentsuch that ι(ϕ) /∈ D. By faithfulness, find sentences σi ∈ SentΛ,S such that JσiK =ι(pi) for 1 ≤ i ≤ n. Let T be any translation such that T (pi) = σi for 1 ≤ i ≤ n.Then JT (ϕ)K = ι(ϕ) /∈ D, and hence T witnesses that ϕ /∈ L(J·K, D). (Cf. Passmann,2018, Proposition 2.50, for a proof in the more general setting for classes of Heytingstructures.) �

Note that the notions of faithfulness and loyalty crucially depend on the choiceof S. As mentioned above, if S′ ⊆ S and J·K is an A-valued S-structure, thenJ·K′ := J·K�SentΛ,S′ is an A-valued S′-structure. Since SentΛ,S′ ⊆ SentΛ,S , we havethat if J·K′ is faithful to A, then so is J·K.

Constructing illoyal models of set theory 5

∀x∀y[∀z(z ∈ x↔ z ∈ y)→ x = y] (Extensionality)

∀x∀y∃z∀w(w ∈ z ↔ (w = x ∨ w = y)) (Pairing)

∃x[∃y(∀z(z ∈ y → 0) ∧ y ∈ x) ∧ ∀w ∈ x∃u ∈ x(w ∈ u)] (Infinity)

∀x∃y∀z(z ∈ y ↔ ∃w ∈ x(z ∈ x)) (Union)

∀x∃y∀z(z ∈ y ↔ ∀w ∈ z(w ∈ x)) (Power Set)

∀p0 · · · ∀pn∀x∃y∀z(z ∈ y ↔ z ∈ x ∧ ϕ(z, p0, . . . , pn)) (Separationϕ)

∀p0 · · · ∀pn−1∀x[∀y ∈ x∃zϕ(y, z, p0, . . . , pn−1)

→ ∃w∀v ∈ x∃u ∈ w ϕ(v, u, p0, . . . , pn−1)] (Collectionϕ)

∀p0 · · · ∀pn∀x[∀y ∈ x ϕ(y, p0, . . . , pn)→ ϕ(x, p0, . . . , pn)]

→ ∀zϕ(z, p0, . . . , pn) (Set Inductionϕ)

Fig. 1. The axioms of ZF formulated in L{∧,∨,→,0,1},{∈}.

However, faithfulness ties J·K very closely to the algebra A: in particular, it cannothold if the algebra A is bigger than the set SentΛ,S , so for countable languages, noA-valued S-structure can be faithful to an uncountable algebra A.

Thus, if A is an uncountable algebra, S an uncountable set of non-logical symbols,J·K is an A-valued S-structure that is faithful to A, and S′ is a countable subset ofS, then J·K′ := J·K�LΛ,S′ cannot be faithful to A.

§3. Algebra-valued models of set theory In the following, we give anoverview of general construction of an algebra-valued model of set theory followingLowe and Tarafder (2015). The original ideas go back to Boolean-valued modelsindependently discovered by Solovay and Vopenka (1965) and were further gener-alised to other classes of algebras (Grayson, 1979; Ozawa, 2007, 2009; Takeuti andTitani, 1992; Titani, 1999; Titani and Kozawa, 2003). Details can be found in (Bell,2005).

In the following, we shall use the phrase “V is a model of set theory” to meanthat V is a transitive set such that (V,∈) |= ZF. Of course, the existence of sets likethis cannot be proved in ZF and requires some (mild) additional metamathematicalassumptions. The choice of ZF as the set theory in our base model is not relevantfor the constructions of this paper and one can generalise the results to models ofweaker or alternative set theories; however, we shall not explore this route in thispaper.

Since we are sometimes working in languages without negation, we need toformulate the axioms of ZF in a negation-free context given in Figure 1, followingLowe and Tarafder (2015, § 3).2 If V is a model of set theory and A is any set, then

2 We should like to stress that the negation-free axioms given are classically equivalentto what is usually called ZF, but not exactly the same axioms: e.g., we use Collectionand Set Induction in lieu of Replacement and Foundation. Many authors call this axiomsystem IZF.

6 B. Lowe, R. Passmann, S. Tarafder

we construct a universe of names by transfinite recursion:

Nameα(V,A) := {x ; x is a function and ran(x) ⊆ Aand there is ξ < α with dom(x) ⊆ Nameξ(V,A)} and

Name(V,A) := {x ; ∃α(x ∈ Nameα(V,A))}.

We let SV,A be the set of non-logical symbols consisting of the binary relationsymbol ∈ and a constant symbol for every name in Name(V,A) (as usual, we usethe name itself as the constant symbol). The language LΛ,SV,A is usually called theforcing language.

If A is a Λ-algebra with underlying set A, we can now define a map J·KA assigningto each ϕ ∈ LΛ,SV,A a truth value in A by recursion (the definition of Ju ∈ vKA

and Ju = vKA is recursion on the hierarchy of names; the rest is a recursion on thecomplexity of ϕ):

J0KA = 0,

J1KA = 1,

Ju ∈ vKA =∨

x∈dom(v)

(v(x) ∧ Jx = uKA),

Ju = vKA =∧

x∈dom(u)

(u(x)→ Jx ∈ vKA) ∧∧

y∈dom(v)

(v(y)→ Jy ∈ uKA),

Jϕ ∧ ψKA = JϕKA ∧ JψKA,

Jϕ ∨ ψKA = JϕKA ∨ JψKA,

Jϕ→ ψKA = JϕKA → JψKA,

J¬ϕKA = ¬JϕKA

J∀xϕ(x)KA =∧

u∈Name(V,A)

Jϕ(u)KA, and

J∃xϕ(x)KA =∨

u∈Name(V,A)

Jϕ(u)KA.

By construction, it is clear that J·KA is an A-valued SV,A-structure and hence,by restricting it to SentΛ,{∈}, we can consider it as an A-valued {∈}-structure.Usually, it is the restriction to SentΛ,{∈} that set theorists are interested in: as aconsequence, we shall use the notation J·KA to refer to the A-valued {∈}-structureand the notation J·KName

A := J·KA to refer to its extension to SentΛ,SV,A .The results for algebra-valued models of set theory were originally proved for

Boolean algebras, then extended to Heyting algebras:

Theorem 2 If V is a model of set theory, B = (B,∧,∨,→,¬,0,1) is a Booleanalgebra or Heyting algebra, and ϕ is any axiom of ZF, then JϕKB = 1.

Proof. Cf. (Bell, 2005, Theorem 1.33 & pp. 165–166). �

Lemma 3 Let H = (H,∧,∨,→,0,1) be a Heyting algebra and V be a model of settheory. Then J·KName

H is faithful to H (and hence, loyal to (H, D) for every designatedset D on H by Lemma 1).

Constructing illoyal models of set theory 7

∧ 0 1/2 10 0 0 01/2 0 1/2 1/21 0 1/2 1

∨ 0 1/2 10 0 1/2 11/2 1/2 1/2 11 1 1 1

→ 0 1/2 10 1 1 11/2 0 1 11 0 1 1

¬0 11/2 1/21 0

Fig. 2. Connectives for PS3

Proof. Consider u := {(∅, a)} ∈ Name(V,H) and ϕ := ∃x(x ∈ u) ∈ SentΛ,SV,H . Itis easy to check that JϕKName

H = a. �

In order to formulate results for implication algebras, Lowe and Tarafder (2015,p. 197) introduced NFF-ZF, the axiom system of all ZF-axioms where the two axiomschemata are restricted to instances of negation-free formulas. They introduced athree-element algebra PS3 (Lowe and Tarafder, 2015, Fig. 2 & § 6) and proved thefollowing result (for the sake of completeness, we give the definition of PS3 in Figure2):

Theorem 4 If V is a model of set theory and ϕ is any axiom of NFF-ZF, thenJϕKPS3 = 1. Furthermore, J·KPS3 is faithful to PS3 and hence loyal to (A, D) forevery designated set D by Lemma 1.

Proof. Cf. (Lowe and Tarafder, 2015, Corollary 5.2) for the first claim. Lowe andTarafder (2015, Theorem 6.2) give a sentence ϕ ∈ SentΛ,{∈} such that JϕKPS3 = 1/2which establishes faithfulness. �

§4. Automorphisms and algebra-valued models of set theory Givena model of set theory V and any Λ-algebras A and B and a Λ-homomorphismf : A→ B, we can define a map f : Name(V,A)→ Name(V,B) by ∈-recursion via

dom(f(u)) := {f(v) ; v ∈ dom(u)} and

f(u)(f(v)) := f(u(v)).

Proposition 5 Suppose that V is a model of set theory, A and B are completeΛ-algebras and f : A→ B is a complete Λ-isomorphism. Let ϕ ∈ LΛ,{∈} with n freevariables and u1, . . . , un ∈ Name(V,A). Then

f(Jϕ(u1, . . . , un)KA) = Jϕ(f(u1), . . . , f(un))KB.

Proof. For atomic formulas, this is easily proved by induction on the rank of thenames involved. For non-atomic formulas, the claim follows by induction on thecomplexity of the formula (where the quantifier cases need the fact that f is abijection). �

Corollary 6 Suppose that V is a model of set theory, A and B are complete Λ-algebras and f : A→ B is a complete Λ-isomorphism. Let ϕ ∈ SentΛ,{∈}. Then

f(JϕKA) = JϕKB.

Corollary 7 Suppose that V is a model of set theory, A is a complete Λ-algebrawith underlying set A, a ∈ A, and that f : A → A is a complete Λ-automorphismwith f(a) 6= a. Then there is no ϕ ∈ SentΛ,{∈} such that JϕKA = a.

8 B. Lowe, R. Passmann, S. Tarafder

Proof. By Corollary 6, if JϕKA = a, then f(a) = a. �

Proposition 8 If A = (A,∧,∨,0,1) is an atomic distributive lattice and a ∈A\{0,1}, then there is a {∧,∨,¬c,0,1}-automorphism f of A such that f(a) 6= a.

Proof. Note that the assumptions imply that A 6= {0,1} and hence At(A) 6= ∅.By atomicity, every permutation π : At(A) → At(A) induces an automorphism ofA preserving ∧, ∨, ¬c, 0, and 1 by fπ(

∨X) =

∨{π(t) ; t ∈ X} for X ⊆ At(A). Let

a =∨Xa. Since a 6= 0, we have Xa 6= ∅; since a 6= 1, we have Xa 6= At(A). So,

pick t0 ∈ Xa and t1 ∈ At(A)\Xa and let π be the transposition that interchangest0 and t1. Then

t0 ≤∨Xa = a, but

t0 6≤∨{π(t) ; t ∈ Xa} = fπ(

∨Xa) = fπ(a),

whence a 6= fπ(a). �

Corollary 9 If V is a model of set theory, B is an atomic Boolean (implication)algebra with more than two elements, and D is any filter on B, then J·KB is loyal,but not faithful to (B, D).

Proof. By Proposition 8, all elements except for 0 and 1 are moved by someautomorphism of an atomic Boolean (implication) algebra and hence by Corollary7, for each sentence ϕ ∈ LΛ,{∈}, we have that JϕKB ∈ {0,1}. In particular, thismeans that L(J·KB, D) = L({0,1}, {1}) = CPC = L(B, D). �

Clearly, atomicity is not a necessary condition for the conclusion of Corollary9: the Boolean algebra of infinite and co-infinite subsets of N is atomless andhence non-atomic, but every nontrivial element is moved by an automorphism,so Corollary 7 applies. We do not know whether this result extends to Booleanalgebras without this property, e.g., rigid Boolean algebras (cf. van Douwen et al.,1980, § 2):

Question 10 Are there (necessarily countable) Boolean algebras B such that J·KBis faithful to B for some designated set D?

§5. Stretching and twisting the loyalty of Boolean algebras

5.1. What can be considered a negation? In this section, we start froman atomic, complete Boolean algebra B and modify it, to get an algebra A thatgives rise to an illoyal J·KA. The first construction is the well-known constructionof tail extensions of Boolean algebras to obtain a Heyting algebra. The other twoconstructions are negation twists: in these, we interpret B as a Boolean implicationalgebra via the definition a → b := ¬a ∨ b, and then add a new, twisted negationto it that changes its logic.

When twisting the negation, we need to pay attention to the fact that not everyunary function on an implication algebra is a sensible negation. In his survey ofvarieties of negation, Dunn (1995) lists Hazen’s subminimal negation as the bottomof his Kite of Negations: only the rule of contraposition, i.e., a ≤ b implies ¬b ≤ ¬a,is required. In the following, we shall use this as a necessary requirement to be areasonable candidate for negation.

Constructing illoyal models of set theory 9

5.2. Tail stretches Let B = (B,∧,∨,→,¬,0,1) be a Boolean algebra, and1∗ /∈ B be an additional element that we add to the top of B to form the tailstretch H as follows: H := B∪{1∗}, the complete lattice structure of H is the ordersum of B and the one element lattice {1∗}, and →∗ is defined as follows:3

a→∗ b :=

a→ b if a, b ∈ B such that a 6≤ b,1∗ if a, b ∈ B with a ≤ b or if b = 1∗,b if a = 1∗.

Lemma 11 The tail stretch H = (H,∧,∨,→∗,0,1∗) is a Heyting algebra withp ∨ ¬p /∈ L(H, {1∗}), so in particular, L(H, {1∗}) 6= CPC.

Proof. If b 6= 0 ∈ B, then by definition b→∗ 0 = ¬b where ¬ refers to the negationin B. In particular, b ∨ ¬Hb = b ∨ ¬b = 1 6= 1∗. �

Lemma 12 If f : B → B is an automorphism of the Boolean algebra B, thenf∗ : H → H defined by

f∗(b) :=

{f(b) if b ∈ B and1∗ if b = 1∗.

is an automorphism of H.

Proof. Easy to check. �

Theorem 13 Let V be a model of set theory, B an atomic Boolean algebra withmore than two elements, and H be the tail stretch of B as defined above. Then theH-valued model of set theory V H is not faithful to H. Furthermore, we have that

(p→ q) ∨ (q → p) ∈ L(J·KH, {1∗})\L(H, {1∗}).

Consequently, V H is illoyal to (H, {1∗}).

Proof. Since B is atomic with more than two elements, each of the non-trivialelements of B is moved by an automorphism of B by Proposition 8. By Lemma 12,these remain automorphisms of H. As a consequence, we can apply Corollary 6 toget that ran(J·KH) ⊆ {0,1,1∗} which is isomorphic to the linear Heyting algebra 3and thus the range is a linear Heyting algebra. As mentioned, Horn (1969) provedthat (p → q) ∨ (q → p) characterises the variety generated by the linear Heytingalgebras, so (p→ q)∨ (q → p) ∈ L(J·KH, {1∗}). However, since B has more than twoelements, we can pick imcomparable a, b ∈ B. Then a →∗ b and b →∗ a are bothelements of B, and thus (p→ q) ∨ (q → p) /∈ L(H, {1∗}). �

We remark that ran(J·KH) = {0,1,1∗}, as can be seen by checking that the J·KH-value of the sentence formalising the statement “every subset of {∅} is either ∅ or{∅}” is 1.

3 In H, we use the (Heyting algebra) definition ¬Hh := h→∗ 0 to define a negation; notethat if 0 6= b ∈ B, ¬Hb = ¬b, but ¬H0 = 1∗ 6= 1 = ¬0.

10 B. Lowe, R. Passmann, S. Tarafder

5.3. Transposition twists Let B = (B,∧,∨,→,¬,0,1) be an atomic Booleanalgebra, a, b ∈ At(B) with a 6= b, and π be the transposition that transposes a andb. Since B is an atomic Boolean algebra, ¬ = ¬c. Then fπ as defined in the proof ofProposition 8 is a {∧,∨,→,¬,0,1}-automorphism of B. We now define a twistednegation by

¬π(∨X) :=

∨{π(t) ∈ At(B) ; t /∈ X}

and let the π-twist of B be Bπ := (B,∧,∨,→,¬π,0,1).4 We observe that the twistednegation ¬π satisfies the rule of contraposition.

Lemma 14 If either ¬ca =∨{t ∈ At(B) ; t 6= a} or ¬cb =

∨{t ∈ At(B) ; t 6= b} is

not in D, then ¬(p ∧ ¬p) /∈ L(Bπ, D). In particular, L(Bπ, D) 6= CPC.

Proof. Without loss of generality,∨{t ∈ At(B) ; t 6= b} = ¬cb = ¬πa /∈ D. Since

a ≤ ¬πa, we have that a = ¬πa ∧ a, and so ¬π(¬πa ∧ a) = ¬πa /∈ D. �

Lemma 15 There is an automorphism f of Bπ such that f(a) = b. In particular,J·KBπ is not faithful to Bπ.

Proof. We know that fπ is an automorphism of B. Since π is a transposition, wehave that π2 = id and π = π−1; using this, we observe that fπ still preserves ¬π:

fπ(¬π(∨X)) = fπ(

∨{π(t) ∈ At(B) ; t /∈ X})

=∨{π(π(t)) ∈ At(B) ; t /∈ X}

=∨{t ∈ At(B) ; t /∈ X}

= ¬π(∨{π(t) ∈ At(B) ; t /∈ X})

= ¬π(fπ(∨X))).

Thus, fπ is an automorphism of Bπ; clearly, fπ(a) = b. The second claim followsfrom Corollary 7. �

Now let V be a model of set theory and J·KBπ the Bπ-valued {∈}-structure derivedfrom V and B.

Lemma 16 If x ∈ ran(J·KBπ ), then ¬πx = ¬cx.

Proof. Let x =∨X for some X ⊆ At(B). By Corollary 7 and Lemma 15, if

x ∈ ran(J·KBπ ), then fπ(x) = x. This means that either both a, b ∈ X or botha, b /∈ X. In both cases, it is easily seen that ¬πx = ¬cx. �

Theorem 17 For any filter D, L(J·KBπ , D) = CPC. In particular, if either ¬ca or¬cb is not in D, then J·KBπ is not loyal to (Bπ, D).

Proof. As mentioned in §2., if we let C := BJ·KBπ = (ran(J·KBπ ,∧,∨,→,¬π,0,1),then L(J·KBπ , D) = L(C, D). But Lemma 16 implies that C = (ran(J·KBπ ,∧,∨,→,¬c,0,1) which is a Boolean algebra (as a subalgebra of B). Thus, L(J·KBπ , D) =L(C, D) = CPC. The second claim follows from Lemma 14. �

4 Note that we do not twist the implication → which remains the implication of theoriginal Boolean algebra B defined by x→ y := ¬cx ∨ y.

Constructing illoyal models of set theory 11

1OO

��

1OO

��L oo // R π // L-- R rr

B 0 0 Bπ

Fig. 3. The four-element Boolean algebra and its transposition twist. Negations areindicated by arrows.

As the simplest possible special case, we can consider the Boolean algebra Bgenerated by two atoms L and R; then, there is one nontrivial transposition π(L) =R and all nontrivial elements of B are moved by the automorphism fπ. As aconsequence of Corollary 7, all sentences will get either value 0 or value 1 underJ·KBπ , and hence L(J·KBπ , D) is classical (cf. Figure 3).

5.4. Maximal twists Again, let B = (B,∧,∨,→,¬,0,1) be an atomic Booleanalgebra with more than two elements and define the maximal negation by

¬mb :=

{1 if b 6= 1 and0 if b = 1

for every b ∈ B. We let the maximal twist of B be Bm := (B,∧,∨,→,¬m,0,1); oncemore observe that the maximal negation ¬m satisfies the rule of contraposition.

Lemma 18 If there is some 0 6= b /∈ D, then (p ∧ ¬p) → q /∈ L(Bm, D). Inparticular, L(Bm, D) 6= CPC.

Proof. Let c := ¬cb. Note that the assumption b 6= 0 implies c 6= 1. In particular,¬mc = 1, and thus c ∧ ¬mc = c. Also

c→ b = ¬cb→ b

= ¬c¬cb ∨ b= b ∨ b = b.

Thus, the assignment ι with p 7→ c and q 7→ b yields ι((p ∧ ¬p)→ q) = b /∈ D. �

Lemma 19 For any b /∈ {0,1}, there is an automorphism f of Bm such thatf(b) 6= b. In particular, J·KBm

is not faithful to Bm.

Proof. We claim that any automorphism f of B also preserves ¬m. Suppose f is anautomorphism of B. If b = 1, then clearly f(¬m1) = f(0) = 0 = ¬m1 = ¬mf(1).Now let b 6= 1. Since f is bijective and f(1) = 1, we have that f(b) 6= 1. Sof(¬mb) = f(1) = 1 = ¬mf(b). The second claim follows from Corollary 7. �

Theorem 20 For any designated set D, L(J·KBm , D) = CPC. In particular, J·KBm

is not loyal to (Bm, D).

Proof. Lemma 19 gives us that every nontrivial element of B is moved by anautomorphism, so we can apply the argument from the proof of Corollary 9: sincefor each ϕ ∈ LΛ,{∈}, we have that JϕKBm

∈ {0,1}, we get that

L(J·KBm , D) = L({0,1}, {1}) = CPC.

The second claim follows from Lemma 18. �

12 B. Lowe, R. Passmann, S. Tarafder

As mentioned at the end of §2., our examples show that restricting the languagecan change faithful models into illoyal ones: for our twisted algebras Bπ and Bm,the general faithfulness result Lemma 3 holds for J·KName

Bπ and J·KNameBm

. However,Theorems 17 & 20 show that their restrictions J·KBπ and J·KBm

are neither faithfulnor loyal.

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