A conjecture about the interpretation of classical mathematics in na¨ıve set theory. Nick Thomas February 1, 2014 Abstract I develop an informal hypothesis stating that there is no natural system of logic in which na¨ıve set theory can interpret a significant fragment of classical mathematics. I formalize this hypothesis as a mathematical conjecture, by first developing a theory of what it means for one formal theory to interpret another formal theory, and operationalizing the notion of a “natural system of logic” by drawing on the previously developed theory of “generalized Routley-Meyer logics.” I then state the conjecture that there is no generalized Routley-Meyer logic in which na¨ıve set theory can interpret Σ01 arithmetic.

A “na¨ıve set theory” is a set theory in which every formula φ(x) defines a set {x : φ(x)}. Na¨ıve set theory is inconsistent in classical logic by Russell’s paradox, but it has been argued, e.g. in Priest [1979], that the comprehension principle endorsed by na¨ıve set theory is sufficiently intuitively compelling that we should abandon classical logic in favor of a logic in which na¨ıve set theory is a coherent theory. Here “coherence” does not necessarily mean consistency, but at least means nontriviality, in the sense that there should be some non-provable statements. Thus it is common (see, e.g., Brady [2006], Restall [1992], Weber [2010]) to study na¨ıve set theory in the context of a paraconsistent logic, where the notions of inconsistency (proving a contradiction) and triviality (proving everything) come apart. Na¨ıve set theory is generally taken to be the following theory, modulo minor variations: Extensionality. Sets with the same members are indistinguishable. ∀x, y(∀z(z ∈ x ↔ z ∈ y) → ∀z(x ∈ z ↔ y ∈ z)). Comprehension. Every formula defines a set. That is, if φ is a formula with X not free, ∃X∀z(z ∈ X ↔ φ). Henceforth we shall mean this theory when we say “na¨ıve set theory,” and we shall also call it NS for short. When we speak of “a na¨ıve set theory,” we mean 1

this theory closed under the entailment relation of some logic. There are logics in which na¨ıve set theory is nontrivial; e.g. LP [Restall, 1992] and DKQ [Brady, 1989, 2006]. However, we generally do not think that nontriviality is a sufficient criterion for a na¨ıve set theory to be a satisfactory theory of sets. We also think that it should be able to do mathematics. Thomas [201+] essentially showed that na¨ıve set theory in LP cannot do mathematics. It is less clear whether na¨ıve set theory in DKQ can do mathematics. Weber [2010] has apparently constructed the natural numbers in this theory. However, key statements such as the statement that successors exist are proven modulo a conjecture, so that whether this construction works remains an open question. Our main goal in this paper is to explore the following informal Hypothesis. There is no natural system of logic in which na¨ıve set theory can do mathematics. By a “natural” system of logic, I mean one which has the properties we usually expect of logics: that it is topic-neutral; it has a recursively enumerable consequence relation; it satisfies the compactness theorem; and so forth. The Hypothesis I have stated is vague, and the goal of this paper is to make a precise mathematical conjecture which expresses at least much of what the Hypothesis is trying to say. Before we turn to that task, though, let us say some words on why we might believe the Hypothesis. There is an argument by inductive reasoning. The body of publications on the subject does not adequately reflect the great deal of time that has been spent looking for a way of reconstructing mathematics in na¨ıve set theory. The reason this effort is not reflected in the publication record is that most of these attempts were not successful. The author is only aware of one serious, published attempt to construct basic mathematics in na¨ıve set theory, which is Weber [2010]; and as stated before it is open whether this construction succeeds, despite its impressive and delicate ingenuity. The point is that many routes to constructing mathematics in na¨ıve set theory have been tried, and so far they have not been definitively successful. Now I give a further heuristic argument for believing the Hypothesis. Any logic in which na¨ıve set theory is nontrivial must sacrifice some of the logical principles needed to prove the basic properties of ordinal numbers used in classical set theory: properties such as that the ordinals are linearly ordered, that successors of ordinals are ordinals, that induction on ordinals works, etc. If all classical facts about ordinals went through unchanged, then the Burali-Forti paradox would trivialize the theory. The ordinals of a na¨ıve set theory could have a classical structure (being linearly ordered, having successors, etc.) up to a certain point, but at some point some features of the classical ordinals must collapse, since the ordinals must behave non-classically by the time we reach the largest ordinal. However, the topic neutrality of logic should mean that whatever principles fail when we reach the largest ordinal, could have failed much lower down. A topic-neutral logic shouldn’t be able to constrain the ordinal structure to 2

collapse only when we reach a very high point in the ordinal structure, because the same logical principles are being used to establish the properties of the ordinals at every level in the ordinal hierarchy. So there should be models of na¨ıve set theory in which the classicality of the ordinal structure collapses at a very early point. In particular, it ought to be the case that it could collapse at ω, so that the finite ordinals failed to have a property such as being linearly ordered, satisfying induction, etc. This would mean that the finite ordinals were not a satisfactory definition of the natural numbers in any given na¨ıve set theory. I argue (still heuristically) that in this case, there would also be no satisfactory definition of the natural numbers in any given na¨ıve set theory. My thinking is that again, by the topic neutrality of logic, if there were logical principles strong enough to construct an infinite set that was linearly ordered, satisfied induction, etc., in every model of the theory, then those same logical principles could have been used to constrain the finite ordinals to have the desired structure. That concludes my informal argument for the Hypothesis. The truth of the Hypothesis would not, in my view, imply that the program of na¨ıve set theory was hopeless. Even if the Hypothesis were true, we could still recapture classical mathematics in the context of a na¨ıve set theory by adding additional axioms and rules which result in the needed logical laws holding in a special portion of the universe which behaves (more or less) classically; see e.g. Thomas [2013a] for this kind of approach. What the Hypothesis would imply would be that the axioms of na¨ıve set theory alone are not enough to yield mathematics. We can have a set universe in which mathematics can be done and unrestricted comprehension is true; but under the Hypothesis, the axioms of na¨ıve set theory would not suffice to constrain the space of models to consist of such universes. It remains to give a mathematical conjecture which expresses the Hypothesis. The rest of the paper focuses on this.

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Theories and theory pairs

We will consider theories in the usual language of first-order logic, with the connectives ∧, ∨, →, ⊥ and the quantifiers ∀, ∃. We define first-order signatures in the usual way, for convenience restricting to signatures containing only relation symbols. We will use a bilateralist conception of what a theory is. In classical logic, to assert ¬φ is the same as denying φ. As pointed out for example in Restall, in paraconsistent contexts this relation can come apart; one can assert φ without denying ¬φ. Moreover, for example in LP, there is no statement whose assertion forces us to deny any other statement, since there is a trivial model in which all statements hold. In LP the act of denial is impossible to reduce to an act of assertion. Classically, a theory is a set of formulas, and we understand that to accept the theory is to assert all of the formulas in the set. In contexts where the

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denial of a formula cannot in general be achieved indirectly by the assertion of a different formula, it is useful to think of a theory as a set of formulas which we assert, together with another set of formulas which we deny. This was suggested in Restall. In line with this, instead of theories, we work with theory pairs (T, T 0 ) (with T, T 0 in a common signature Σ), where we understand T as the set of formulas we assert, and T 0 as the set of formulas we deny. These theory pairs are different from Restall’s “bitheories,” in that Restall’s bitheories are considered to be set in some system of logic and to be closed under logical consequence, whereas we have no need for this stipulation. We will, however, tend to follow the convention that when a theory pair (T, T 0 ) is associated with a system of logic, T is closed under the consequence relation of that logic, and T 0 is the set of formulas φ such that T, φ ` ψ for all ψ; i.e., T 0 is the set of formulas which lead to triviality in conjunction with T . We say that a theory pair (T, T 0 ) is “coherent” iff T ∩ T 0 = ∅; i.e., there is no statement which we both assert and deny.

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Translations and the interpretation relation

We want to formalize the idea that a given na¨ıve set theory “does mathematics.” By this we must mean that the na¨ıve set theory is able to interpret some classical mathematical theory. In other words, it should be able to assert at least some substantial portion of the classical theory’s theorems, and it should be able to deny at least some substantial portion of the theorems refutable in the classical theory. Obviously the theorems do not have to take a syntactically identical form, but rather can be stated under some appropriate translation. Let us begin with this notion of an “appropriate translation.” Let Σ, ∆ be signatures. A “translation” from Σ to ∆ is a tuple tr = (κ(x), {ρR (x1 , ..., xn )}R∈Σ

an n-ary relation symbol ),

where κ(x), ρR (x1 , ..., xn ) are formulas in ∆ with all free variables shown. Letting tr be a translation as shown, we define a function from formulas in Σ to formulas in ∆, which we also denote tr(φ), by the recursive clauses tr(R(x1 , ..., xn )) = ρR (x1 , ..., xn ), tr(¬φ) = ¬tr(φ), tr(φ ∧ ψ) = tr(φ) ∧ tr(ψ), tr(φ ∨ ψ) = tr(φ) ∨ tr(ψ), tr(φ → ψ) = tr(φ) → tr(ψ), tr(∀x(φ)) = ∀x(κ(x) → tr(φ)), tr(∃x(φ)) = ∃x(κ(x) ∧ tr(φ)).

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The translations we consider work by restricting quantifiers to some definable class, and translating atomic relations as arbitrary definable predicates over the definable class. For example, let Σ be the language of arithmetic, and ∆ the language of set theory. For the usual interpretation of arithmetic in classical set theory, we would let κ(x) be the statement “x is a finite von Neumann ordinal,” and we would interpret the relations S(x, y), +(x, y, z), ·(x, y, z), < (x, y) by the apdef

propriate relations on the von Neumann ordinals: ρS (x, y) = (y = x ∪ {x}), etc. We have the notion of a translation; now we need to define the notion of one theory (pair) interpreting another theory (pair) under a given translation. Let (T, T 0 ) be a theory pair in the signature Σ, (U, U 0 ) a theory pair in the signature ∆, and tr a translation from Σ to ∆. We say that “(U, U 0 ) interprets (T, T 0 ) under the translation tr” iff for all formulas φ in Σ: 1. If φ ∈ T , then tr(φ) ∈ U . 2. If φ ∈ T 0 , then tr(φ) ∈ U 0 . Finally, we say that “(U, U 0 ) interprets (T, T 0 )” iff there is a translation tr from Σ to ∆ such that (U, U 0 ) interprets (T, T 0 ) under tr.

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What theories of mathematics can na¨ıve set theory not interpret?

Let (T, T 0 ) be a na¨ıve set theory. It will be convenient to assume (T, T 0 ) is recursively enumerable. We want to say that there is no theory pair (U, U 0 ) representing a substantial fraction of classical mathematics which (T, T 0 ) can interpret. A very modest conjecture would say that (T, T 0 ) cannot interpret ZFC. In fact, there is a significant class of na¨ıve set theories for which a weak analogue of this conjecture is easy to prove. Theorem 1. Let U be a consistent, recursively enumerable classical theory containing basic arithmetic, such that U ` ((T, T 0 ) is coherent). Then U 0 ((T, T 0 ) interprets U ). Proof. Suppose it were provable in U that (T, T 0 ) interprets U . Within U , we will prove that U is consistent, so that by G¨odel’s theorem it is inconsistent, contradicting our assumption. Within U , argue as follows. (T, T 0 ) interprets U , so let tr be a translation witnessing this. Suppose U ` φ ∧ ¬φ. Then U ` φ. Then tr(φ) ∈ T . U ` ¬φ, so tr(φ) ∈ / T . This is a contradiction, so U 0 φ ∧ ¬φ. That is, U is consistent. This shows, for example, that it is not provable in ZFC (if ZFC is consistent) that na¨ıve set theory in DKQ — the theory (T, T 0 ) where T = {φ : NS `DKQ φ}, 5

T 0 = {φ : ∀ψ(NS, φ `DKQ ψ)} — interprets ZFC. This is true because the nontriviality of NS in DKQ, and therefore the coherence of (T, T 0 ), is provable in ZFC. [Brady, 1989] The same argument shows that it is not provable in second-order arithmetic that na¨ıve set theory in DKQ interprets second-order arithmetic, since the model construction of Brady [1989] can be carried out in second-order arithmetic. Theorem 1 is quite unsurprising: it essentially says that the more classical mathematics a theory pair can interpret, the harder it is to prove it coherent. This is a natural outgrowth of G¨odel’s theorem. Unfortunately, Theorem 1 will not give us any statements of the form “(T, T 0 ) does not interpret U ;” it only tells us conditions under which “(T, T 0 ) interprets U ” is unprovable in U . This is the first limitation of the theorem, with respect to our project. The second limitation is that the only classical theories whose interpretability in (T, T 0 ) Theorem 1 will tell us anything about are those theories which are strong enough prove (T, T 0 ) coherent. But I think that in most cases, a na¨ıve set theory (T, T 0 ) can actually interpret substantially less mathematics than is needed to prove it coherent. For these reasons, I think an entirely different approach to the question is needed than that given by Theorem 1. We want to say that (T, T 0 ) cannot interpret any substantial fraction of classical mathematics. To make this conjecture precise, we should pick a specific theory (U, U 0 ) representing some weak fragment of classical mathematics, and say that (T, T 0 ) cannot interpret it. For this purpose, we will pick Σ01 arithmetic. By “Σ01 arithmetic,” we mean the theory pair (U, U 0 ) in the language of arithmetic where U = {φ : φ is Σ01 and true}; U 0 = {φ : φ is Π01 and false}. U and U 0 are both recursively enumerable. All formulas of U are provable in Robinson arithmetic, and all formulas of U 0 are refutable in Robinson arithmetic, because Robinson arithmetic is Σ01 -complete, and a negated Π01 formula is equivalent to a Σ01 formula. Σ01 arithmetic is a very weak fragment of mathematics, but it is enough, for example, to guarantee that no theory pair interpreting it is decidable (because Σ01 arithmetic is undecidable). I think it is uninterpretable in na¨ıve set theory because, despite being highly finitary, it is still slightly infinitary in that it involves unbounded quantifiers, whose interpretation requires the existence of a definable class sufficiently like the natural numbers. Moreover, despite not including any instances of the axiom of induction, interpreting it nonetheless seems to require that some induction-like reasoning be possible in the interpreting theory, because defining the operations + and · in the usual way uses induction.

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Generalized Routley-Meyer logics

We have answered the question of what theory we wish to say that na¨ıve set theory cannot interpret in any natural logic. Now we need to say what we mean by a “natural” logic. For this purpose we will use the theory of “generalized Routley-Meyer (GRM) logics” introduced in Thomas [2013b]. This theory axiomatizes a class of logics which are definable by model theories in a generalization of Routley-Meyer style semantics. GRM logics satisfy a number of desirable properties. They are all subclassical; they all have recursively enumerable consequence relations; they all satisfy the compactness theorem; and they all satisfy the standard structural and conjunction/disjunction introduction/elimination rules. Moreover, they are topic-neutral, in the sense that GRM logics cannot constrain models to treat any given non-logical symbol in a special way. To the author’s knowledge, all well-known examples of logics satisfying the properties just listed are GRM logics. This means that GRM logics are at once a fairly general framework, and one that is sufficiently narrow that it specifies logics with a lot of natural, desirable properties. A GRM logic is specified by a class of (generalized) Routley-Meyer frames, which are used to generate the class of valid models of the logic. We think of these frames as themselves being models in classical first-order logic, and we let the class of frames of a GRM logic be axiomatized by a first-order theory. This restriction, that the class of frames be first-order axiomatizable, is what makes it true that GRM logics are compact and recursively enumerable. The semantics for weak relevant logics (as given by Kit Fine in §53 of Anderson et al. [1992]) requires that the logic be able to control, in fairly elaborate ways, the domains of the points in the models of its frames. To satisfy this need, we take the unconventional step of making the domains of points part of the frame itself. Now we define GRM logics, following Thomas [2013b]. We begin by defining the concept of a “point frame,” which is a frame without the domains of points included (so in essence, a frame in the usual sense). A point frame, like a frame, is a kind of model in classical first-order logic. The signature of point frames is ΣP = {R, N, ≤∀ , ≤∃ , ω}, where R is a ternary relation symbol, N, ≤∀ , and ≤∃ are binary relation symbols, and ω is a constant symbol. (Note that we will always denote points by Greek letters, because we wish to reserve Roman letters for the objects of domains.) N generalizes the Routley star; it is like the Routley star, but not required to be a function. In frames where for every α there is a unique β such that αN β, we continue to denote that unique β by α? . In general for α to satisfy ¬φ, we will require that φ be false at all points β such that αN β. Our generalization from ? to N is required to accommodate intuitionistic logic under its usual semantics, and also to accommodate relevant logics under Fine’s semantics. ≤∀ and ≤∃ say what worlds we look at to evaluate quantified statements; for

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instance, in intuitionistic logic we must look at all worlds later in the accessibility order to evaluate universal quantifiers. ω is the “real world,” usually called 0 in relevance logic literature. We use the Greek letter ω in order to follow our convention that points are denoted by Greek letters. Definition 1. A “point frame” P is a (first-order, classical) model in the signature ΣP . We require it to satisfy the following first-order statements: 1. (Negation-worlds exist.) ∀α∃β(αN β). 2. (Counterfactual worlds exist.) ∀α∃β, γ(Rαβγ). 3. (Quantifiers range over something.) ∀α(∃β(α ≤∀ β) ∧ ∃β(α ≤∃ β)). The reason for axioms (1)-(3) is that in their absence, we could construct logics which were not subclassical; see Thomas [2013b] for details. Now let us turn to full frames, which consist of points and the domains of the points. For the language of frames we will use two-sorted first-order logic, with the two sorts being called “points” and “objects.” For variables denoting points we will use Greek letters α, β, ..., and for variables denoting objects we will use Roman letters a, b, .... The signature of frames is ΣF = ΣP ∪ {∈, 7→}, where: 1. The relation symbols in ΣP range over points, and ω ∈ ΣP is a point constant symbol. 2. ∈ is a binary relation symbol taking an object on the LHS and a point on the RHS. 3. 7→ is a quaternary relation symbol taking an object, a point, an object, and a point, in that order. We write (a, α) 7→ (b, β) as syntactic sugar for 7→(a, α, b, β). a ∈ α is meant to be read as, “the object a is part of the domain of α.” The statement (a, α) 7→ (b, β) means that object a at world α is a “modal ancestor” of object b at world β. All atomic statements true about a at α become true of b at world β. This is a more fine-grained version of the usual monotonicity relation ≤ between points. This more fine-grained version is used in relevant logics under Fine’s semantics. Definition 2. A “frame” F is a two-sorted model of (possibly an expansion of) ΣF . We require, furthermore, that F satisfy the following statements: 1. (Domains are nonempty.) ∀α∃a(a ∈ α). 2. (7→ is reflexive.) ∀α∀a((a, α) 7→ (a, α)).

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If F is a frame, we let Fpt denote F ’s set of points, and Fob denote F ’s set of objects. Given α ∈ Fpt , we define Fob (α) = {a ∈ Fob : F |= a ∈ α}. Definition 3. Let F be a frame, and Σ a first-order signature, which may contain relation and constant symbols. An “F, Σ-model” is a tuple M = (F, I, J), where: 1. I is a function defined on pairs (α, R) of points in F and relation symbols in Σ, such that for each α ∈ Fpt and each n-ary relation R ∈ Σ, I(α, R) ⊆ (Fob )n . 2. J is a function defined on the constant symbols in Σ such that for each c ∈ Σ, J(c) ∈ Fob . Furthermore, we require that: 1. For all constant symbols c ∈ Σ and all α ∈ Fpt , F |= J(c) ∈ α. 2. For all α, β ∈ Fpt , all n-ary relation symbols R ∈ Σ, and all a1 , ..., an , b1 , ..., bn ∈ Fob , if for each 1 ≤ i ≤ n, F |= ai ∈ α, bi ∈ β, (ai , α) 7→ (bi , β), then if (a1 , ..., an ) ∈ I(α, R) then (b1 , ..., bn ) ∈ I(β, R). Let M = (F, I, J) be an F, Σ-model. We define the satisfaction relation |=M (or just |= where clear) between elements α of Fpt and formulas φ in Σ without free variables, which may also mention elements of Fob . It is defined as follows: 1. α 2 ⊥. def

2. α |= R(t1 , ..., tn ) iff (δM (t1 ), ..., δM (tn )) ∈ I(α, R), where δM (ti ) = J(ti ) def

if ti is a constant symbol in Σ, and δM (ti ) = ti if ti ∈ Fob . 3. α |= φ ∧ ψ iff α |= φ and α |= ψ. 4. α |= φ ∨ ψ iff α |= φ or α |= ψ. 5. α |= ¬φ iff for all β ∈ Fpt such that F |= αN β, β 2 φ. 6. α |= φ → ψ iff for all β, γ ∈ Fpt such that F |= Rαβγ, if β |= φ then γ |= ψ. 7. α |= ∀x(φ(x)) iff for all β ∈ Fpt such that F |= α ≤∀ β and all a ∈ Fob such that F |= a ∈ β, β |= φ(a). 8. α |= ∃x(φ(x)) iff for some β ∈ Fpt such that F |= α ≤∃ β and some a ∈ Fob such that F |= a ∈ β, β |= φ(a). We say that M |= φ iff ω F |= φ.

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Definition 4. A “generalized Routley-Meyer (GRM) logic” L is a recursively enumerable theory of two-sorted first-order logic in some signature Σ ⊇ ΣF . We require, furthermore, that: 1. (Nontriviality.) There is a frame F such that F |= L. 2. (Domain expansion property.) If f : F → G is a domain contraction and G |= L, then F |= L. Let L be a generalized Routley-Meyer logic, and Σ a first-order signature. An “L, Σ-model” is any F, Σ-model where F |= L. Now let T be a first-order theory in Σ, and φ a formula in Σ (all without free variables). We say that T |=L φ iff for all L, Σ-models M , if M |= ψ for all ψ ∈ T then M |= φ. Both axioms on GRM logics (nontriviality and the domain expansion property) are required for subclassicality. Nontriviality is an obvious requirement, and the domain expansion property is needed to prevent, for example, a logic which entails that in every model ω has no more than five objects in its domain. Again see Thomas [2013b] for details.

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The conjecture

We can now state the conjecture which formalizes the Hypothesis. Conjecture 1. Let L be a generalized Routley-Meyer logic. Then the theory pair NSL = (T, T 0 ) given by T = {φ : NS `L φ}, T 0 = {φ : NS, φ `L ⊥} does not interpret Σ01 arithmetic.

References Alan Ross Anderson, Nuel D. Belnap, and J.M. Dunn. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, 1992. Ross Brady. Universal Logic. University of Chicago Press, 2006. Ross T. Brady. The non-triviality of dialectical set theory. In Paraconsistent Logic: Essays on the Inconsistent. Philosophia Verlag, 1989.

Andres Caicedo. Answer to ’Possible Turing degrees of countable models of ZFC’. On math.stackexchange.com. URL is http://math.stackexchange.com/questions/607393/possible-turing-degrees-of-countable-mod 2013. Kenneth Kunen. Set Theory. College Publications, 2011. 10

Graham Priest. The Logic of Paradox. Journal of Philosophical Logic, 8(1), 1979. Greg Restall. Assertion, denial, and non-classical theories. To appear in the proceedings of the Fourth World Congress of Paraconsistency, Melbourne July 2008. Greg Restall. A note on na¨ıve set theory in LP. Notre Dame Journal of Formal Logic, 33(3), 1992. Nick Thomas. Expressive limitations of na¨ıve set theory in LP and minimally inconsistent LP, 201+. To appear in the Review of Symbolic Logic. Available at https://sites.google.com/a/uconn.edu/nick-thomas/. Nick Thomas. Recapturing classical mathematics paraconsistent set theory, 2013a. Available https://sites.google.com/a/uconn.edu/nick-thomas/.

in at

Nick Thomas. A generalization of the Routley-Meyer semantic framework, 2013b. Under submission at the Journal of Philosophical Logic. Zach Weber. Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic, 3, 2010.

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