Modal Set Theory* Christopher Menzel 1 Modal Set Theory and Traditional Modal Metaphysics Set theory is the study of sets using the tools of contemporary mathematical logic. Modal set theory draws in particular upon contemporary modal logic, the logic of necessity and possibility. One simple and obvious motivation for modal set theory is the fact that, from a realist perspective that takes the existence of sets seriously, sets have philosophically interesting modal properties. For instance, perhaps the most notable and distinctive property of sets is their extensionality: sets a and b are identical if they have exactly the same members; formally, where we take variables from the lower end of the alphabet to range over sets: Ext ∀a∀b(∀x(x ∈ a ↔ x ∈ b) → a = b). Intuitively, however, extensionality is not a contingent matter, a mere matter of happenstance. Rather, there simply couldn't have been distinct sets that shared all their members; there is no such possible world. That is, at a minimum, we want to be able to express that extensionality is a necessary truth: □Ext □∀a∀b(∀x(x ∈ a ↔ x ∈ b) → a = b). Clearly, however, there is much more than this to the modal connection between sets and their members. For note that, for all □Ext tells us, one and the same set a could have vastly different members from one world to the next, so long as it remains the case that, in each world, no other set has exactly the same members as a in that world. Intuitively, however, the intimate connection between a set and its members is maintained across worlds; if a set has Angela Merkel, say, as a member, it could not possibly have failed to have her as a member. Sets, that is to say, have their members essentially; if x is a member of a, then it is a member of a in every world in which a * The author is grateful to Øystein Linnebo, Neil Barton, and John Wigglesworth for helpful and illuminating comments on earlier drafts of this entry. 2 exists, i.e., in every world in which something is identical to a; formally, letting E!t abbreviate ∃y y = t, for terms t: E∈ □∀a∀x(x ∈ a → □(E!a → x ∈ a)), Likewise, non-membership; a set cannot "add" new members in one world that it lacks in another: E∉ □∀a∀x(x ∉ a → □(E!a → x ∉ a)). A related philosophical issue is not settled by the preceding principles. Suppose x is a member of a set a here in the actual world and that a exists in some other possible world w. Then by E∈, x is a member of a in w. But nothing follows about x's existence in w. For all we know from E∈, all sets might exist necessarily, even those that have contingent members, members that might not themselves have existed. Hence, if that is so, the singleton set {Merkel}, for example, would have existed even if Merkel hadn't. However, on most conceptions of set, sets are ontologically dependent upon their members and, hence, could not themselves exist without their members existing; there could be no singleton set {Merkel} without Merkel. On such a conception, then, we are in need of a further modal principle: OD □∀a∀x(x ∈ a → □(E!a → E!x)).1 Given OD and the assumption that Merkel is a contingent being, ◇¬E!m, it now follows that the set {Merkel} too is contingent, as expected; it fails to exist at any Merkel-free possible world. Surprisingly, at first blush anyway, the assumption that there are contingent beings is neither philosophically nor mathematically trivial. Regarding the latter, it is a well-known theorem of the simplest and most straightforward system of modal predicate logic that there neither are nor could have been any contingent beings, i.e., that, necessarily, everything there is exists necessarily: NE □∀x□E!x.2 1 The three principles above are still jointly consistent with the possibility of sets a that contain members x that cannot coexist with a. This (rather perverse) possibility can be ruled out by replacing OD with a principle asserting that the membership relation is existence entailing: □∀a□∀x□(x ∈ a → (E!a ∧ E!x)). Given E∈, both E∉ and OD follow (assuming the propositional modal logic S5). 2 See (Menzel, 2018) for a formal proof of this and other controversial theorems in the simplest modal predicate logic, as well as discussion of the surrounding philosophical issues. 3 Avoiding this consequence requires choosing between restrictions (of varying severity) on one's logical system, each with its own virtues and liabilities.3 However, some philosophers - so-called necessitists - embrace NE (from which, of course, OD trivially follows) and choose instead to offer sophisticated philosophical explanations of the allegedly mistaken naïve intuition that some things might not have existed.4 The choice of a logic and the adoption of a philosophical standpoint about the metaphysics of sets are therefore interestingly interdependent. It is not our purpose here to adjudicate these issues. Rather, the point of this initial section has been to illustrate one powerful motivation for modal set theory, namely, its usefulness as a tool for exploring quite traditional lines of inquiry in modal metaphysics concerning contingency, essentiality, ontological dependence, and the like that surface naturally in connection with the existence of sets.5 The remainder of this article, however, will be devoted to recent development in modal set theory with regard to a rather more directed inquiry into both the nature and structure of sets that is motivated in particular by the attractive prospect of a satisfying explanation of Russell's Paradox. 2 ZF and Russell's Paradox6 The years 1897-1903 saw the emergence of a string of related paradoxes concerning the notions of number, set, class, property, proposition, and truth.7 Among those concerning sets, Russell's Paradox is undoubtedly the best known and arguably the one most directly responsible for subsequent developments in the foundations of mathematics. The argument is well-known. Its heart, of course, is the principle of naïve comprehension, i.e., the principle that, for any property of things, there is the set of things that have that property. More formally (and somewhat anachronistically) expressed in the language of first-order logic, it is the principle that, for any predicate	φ(x), there is the set of things it is true of: 3 The system Q of (Prior, 1968) is the most severely restricted of well-known systems, abandoning in particular the interdefinability of □ and à and most familiar principles of propositional modal logic. For examples of less severely restricted systems, see (Kripke, 1963), (Fine, 1978), and (Menzel, 1991). 4 The term "necessitism" and its cognates was coined in (Williamson, 2010), although the view was in large measure anticipated and developed in detail in (Linsky & Zalta, 1994). See also (Salmon, 1987) for an influential precursor. 5 (Van Cleve, 1985) is a fine exploration of the issues raised in this section. See also (Fine, 1981) for a detailed and rather more formal study. 6 The exposition in this section is similar to that found in Sections 3-6 of (Menzel, 2017), which was written largely in parallel with the current entry, albeit toward very different ends. 7 See (Cantini, 2014) for an excellent overview. 4 NC ∃a∀x(x ∈ a ↔ φ(x)), where 'a' is not free in φ(x). Intuitively, at first sight anyway, the principle seems airtight. For a well-defined predicate unambiguously picks out some existing things (or perhaps no things at all), and what more could you need for the existence of a set than the existence of its purported members, the things that constitute it? For all its intuitive appeal, of course, NC is inconsistent: given the well-defined predicate 'x ∉ x', by NC we have the "Russell set" r containing exactly those things x that are not members of themselves, i.e., R ∀x(x ∈ r ↔ x ∉ x). Instantiating with r, the contradiction that r ∈ r if and only if r ∉ r follows immediately. The best known and most influential response to Russell's Paradox is of course that of Ernst Zermelo. To express Zermelo's ideas, it is useful to speak of mere pluralities of things, where such talk is to be thought of as "ontologically innocent". That is, talk of a plurality of things is not to be understood to refer to some additional thing over and above the things we are talking about - a set or class or mereological sum that they constitute - but, rather, simply as a convenient way of talking about those things jointly, or collectively, as we seem freely to do when we use plural noun phrases in sentences like "It took three men to lift the piano" and "The fans went wild". The lesson of Russell's Paradox, then, in these terms, is that not all pluralities of things can safely be assumed to constitute a further thing, viz., a set that contains them; in particular, to assume without qualification, as NC would have it, that the things an arbitrary predicate is true of constitute a set can be logically catastrophic. At the same time, some pluralities seem clearly safe. Zermelo's brilliantly executed idea (Zermelo, 1908)8 - implemented in his axiomatic set theory Z - was to stipulate the existence of some initial sets to get things going and then introduce a variety of sound "set-building" operations that lead safely from given objects or sets to new sets. We will describe Z in some detail. Zermelo begins with the extensionality axiom Ext. His next axiom, the axiom of elementary sets, is actually a combination of an existence axiom and a set-building axiom. Specifically, he postulates the existence of the empty set ∅, 8 Translated as (Zermelo, 1967); see also the informative introductory note by Felgner that accompanies the translation of this paper in the polyglot edition (Zermelo, 2010) of Zermelo's collected works, pp. 160-89. Zermelo's theory included the important but controversial axiom of Choice, though it will play no part here. 5 ES ∃a∀x x ∉ a, and introduces the axiom of Pairing, which says, in effect, that any pair of (not necessarily distinct) objects x and y are jointly safe and hence constitute a set {x, y}:9 Pr ∃a∀z(z ∈ a ↔ (z = x ∨ z = y)). Assuming extensionality, these two axioms alone already give us the power to prove the existence of the infinite series of Zermelo numbers ∅, {∅}, {{∅}}, {{{∅}}}, ..., so-called because they served as Zermelo's surrogates for the natural numbers. (For convenience, abbreviate them, respectively, as ∅A, ∅B, ∅C, ∅D, ... .) However, in addition to so-called "pure" sets like these that are "built up" solely from the empty set, Zermelo also made room in his theory for the existence of arbitrarily many urelements, that is, things that are not themselves sets - persons, planets, natural numbers, etc. - and, hence, by Pr, for the existence of "impure" sets built up from them. And although he didn't explicitly assume it as an axiom in 1908, it is in the spirit of his theory to take the urelements to constitute a set U of their own:10 Ur ∃a∀x(x ∈ a ↔ (x ≠ ∅ ∧ ∀y y ∉ x)). Let ZU be Zermelo's theory Z with the additional axiom Ur. NC is of course absent from Zermelo's theory but a significant remnant of it remains in the form of a set-building principle, Separation. Given a predicate φ(x), NC called sets into being ex nihilo from the things of which φ(x) is true. Separation, by contrast, only vouches for the things in some previously given set b that φ(x) is true of: Sep ∃a∀x(x ∈ a ↔ (x ∈ b ∧ φ(x)), where 'a' is not free in φ(x).11 Zermelo's next two set-building axioms are Union and Powerset, which tell us, respectively, that, for a given set b, the members of the members of b as well as the subsets of b safely constitute sets of their own: 9 We will make free use of the common {xB, ... , xH} notation for finite sets without defining it formally. 10 The existence of U is not entirely unproblematic, as it could turn out to be inconsistent with the other axioms if there are "too many" urelements. See, e.g., (Nolan, 1996) and (Menzel, 2014). Zermelo himself wasn't sure how to work urelements into his theory until over two decades later; see (Zermelo, 1930) and Kanamori's informative introductory note to its translation in (Zermelo, 2010), pp. 390-430. 11 Sep renders ES otiose, since it is a truth of (classical first-order) logic that something x exists, ∃xE!x, from which ES follows directly from Pr and the instance of Sep where φ(x) is x ≠ x. 6 Un ∃a∀x(x ∈ a ↔ ∃c(c ∈ b ∧ x ∈ c)) Pow ∃a∀x(x ∈ a ↔ x ⊆ b). Although formulated decades before the mature conception of the set theoretic universe in (Zermelo, 1930), these two axioms clearly anticipated it. For, given the initial set U of urelements - call it UA - by Pow the set ℘(UA) of all of its subsets exists. By Pr and Un, we have the set UB = UA ∪ ℘(UA)12 consisting of all the members of UA and ℘(UA). Applying Pow again we have the set ℘(UB) of all the subsets of UB which we can then join with UB itself to yield the set UC = UB ∪ ℘(UB). In general: D1 UA = U UHOB = UH ∪ ℘(UH). Even in Zermelo's early work, then, the sets are naturally taken to have a structure that is cumulative and hierarchical, advancing "upwards" via iterations of the powerset and (binary) union operations, from an initial stock of urelements, in an ever-expanding series of stages, or levels, each successive level UHOB consisting of everything in the preceding level together with all the sets that can be formed from them, as indicated in Figure 1. Say that one level UH is higher than another UQ just in case n > m (equivalently, in light of their cumulative nature, just in case UQ ⊂ UH) and that the level λ(x) of an object x is the first level of the hierarchy in which it occurs. Since (a) we begin with a base set UA of urelements, (b) the hierarchy grows discretely from one level to the next, and (c) a set of level UHOB is always constituted by objects in level UH, it should also be clear that the sets on this conception are all well-founded: no set can be a member of itself and, more generally, there are no infinitely descending membership chains ... ∈ aHOB ∈ aH ∈ ... ∈ aB ∈ aA. Since the axioms above do not explicitly rule out such structural impossibilities, it must be done independently by means of a separate principle; in Z, this is the axiom of Foundation, which requires every nonempty set a to have a member with which it shares no members: 12 a ∪ b is the binary union of sets a and b, and is definable as the set whose existence follows by Un from the set {a, b}, whose existence in turn follows from a and b by Pr. 7 Fnd a ≠ ∅ → ∃y(y ∈ a ∧ ∀z(z ∈ y → z ∉ a)).13 We turn now to the critical Zermelian axiom of Infinity. What is particularly important about this axiom, especially for purposes here, is that it asserts not merely the safety of an infinite plurality - Ur will have already done that on the assumption that there are infinitely many urelements - but, rather, the safety of a plurality that is finitely unbounded, i.e., unbounded (hence infinite) in our hierarchy of finite (i.e., finitely-indexed) levels UH. Consider, in particular, the Zermelo numbers ∅A, ∅B, ∅C, ... . As λ(∅V) = UVOB, for all natural numbers i, it follows that, for every finite level UH, no matter how high, there is a Zermelo number (∅H, for example) that only first occurs in a higher level still; the Zermelo numbers are thus unbounded in the hierarchy of finite levels UA, UB, UC, ... and, hence, never constitute a set in any of them. The Zermelo numbers, then, are of a rather different sort structurally than any we've encountered hitherto. But, ultimately, from the realist's standpoint, at least, there doesn't seem to be any more reason to question their safety than there is to question the safety of the urelements that we sanction in Ur that get the hierarchy going in the first place or the plurality of subsets of a given level that we sanction in Pow that enable us, at any given level, to extend the hierarchy to 13 Zermelo did not include Foundation in his 1908 axiomatization but, as non-well-founded sets were not defined and studied in any systematic way until (Mirimanoff, 1917) and the iterative conception was at most only beginning to take shape in Zermelo's mind, it seems likely that he did not at the time recognize any pressing need for the axiom. Figure 1: The finite levels of the cumulative hierarchy 8 the next level. For, just as in those cases, all of the things in question are there and hence available for collection into a set. Moreover, as the Zermelo numbers are unbounded in the hierarchy of finite levels and hence could not constitute a set at any such level, the set they would constitute would demonstrably not be a member of itself. So the assumption that they are jointly safe does not appear to raise the specter of a Russell-style paradox. Accordingly, Zermelo's Infinity axiom posits the collective safely of the Zermelo numbers the way that Ur does for the urelements and Pow does for the subsets of a given set. More exactly, it declares that there is a set that contains the Zermelo numbers: Inf ∃a(∅ ∈ a ∧ ∀x(x ∈ a → {x} ∈ a)). The set of Zermelo numbers proper - call it ζ (zeta) - can then be derived straightaway by Sep.14 So the axiom of Infinity implies, not just that there are infinite sets, but that there are infinite sets whose members occur at arbitrarily high finite levels of the hierarchy. Hence, intuitively, the cumulative hierarchy UA, UB, UC, ..., of levels must continue beyond the finite. For this to be provable, however, we must first show that they themselves form a set {UA, UB, UC, ... }. It certainly seems that they should, for exactly the same reasons that were cited for the safety of the structurally similar plurality of Zermelo numbers - they are all there and no obvious paradox arises from the assumption that they constitute a set. But Inf and the other axioms of ZU are not enough to guarantee this.15 Thus, a further principle is needed, one typically attributed to Abraham Fraenkel,16 the axiom schema of Replacement, the addition of which to Zermelo's theory Z(U) gives us ZF(U). Replacement captures the structural intuition that if the members of a given set b can be correlated one-to-one with a given plurality, then that plurality also constitutes a set. More formally, where ∃!xφ as usual means that something is uniquely φ: Rep ∀x(x ∈ b → ∃!yψ(x, y)) → ∃a∀y(y ∈ a ↔ ∃x(x ∈ b ∧ ψ(x, y))). 14 Specifically, by letting b be the set given by Inf and letting φ(x)	be	∀y[]∅ ∈ y ∧ ∀z(z ∈ y → {z} ∈ y)^ → x ∈ y], i.e., the predicate "x is in every set y that contains ∅ and the singleton of any of its members". 15 To see this, very briefly: where Va is the set of hereditarily finite pure sets, let Wc = Va ∪ U and WHOB = WH ∪ ℘(WH), for n ∈ N, and let W = ⋃H∈NWH. It is easy to see that W is a model of ZU (recall that VaOa ⊆ W is a model of Z) and that UV ∈ W for all i ∈ N but that {UA, UB, UC, ... } ∉ W. The author thanks Noah Schweber and Joel David Hamkins for this construction. 16 Skolem independently identified the need for Replacement, and his explicitly first-order formulation of the principle is essentially the one that is mostly used today. See (Fraenkel, 1922) and (Skolem, 1922); an English translation of the latter can be found in (van Heijenoort, 1967), pp. 290-301. 9 To see how Rep enables us to show that the finite levels are jointly safe, let L(y) mean that y is a finite levels.17 Let ψ(x, y) be 'x ∈ ζ ∧ L(y) ∧ x ∈ y ∧ ∀z((L(z) ∧ x ∈ z) → y ⊆ z)', i.e., "x is a Zermelo number and y is its level". ψ(x, y) correlates the members of ζ one-to-one with the levels UB, UC, ... and, hence, by Rep, they constitute a set to which (by Pr and Un) we can add the initial level UA of urelements; so all the levels jointly constitute our desired set {UA, UB, UC, ... }. By Un their union is a set, so all the members of all the finite levels do indeed form a set Ua of their own, the first transfinite level of the hierarchy18 - the result, as it were, of putting a "disk" atop the hierarchy of finite levels depicted in Figure 1 indicating its "completion" in a further level. That of course is not the end of the hierarchy but simply a new starting point for iterating the powerset and binary union operations to generate yet further levels UaOB, UaOC, ..., which (by Rep) jointly form a set and hence (by Un) constitute a new limit level UaOa, and thus once again further levels UaOaOB, UaOaOC, ..., and so on, as depicted in Figure 2. In general, then, by including a limit clause representing the continual "completion" of these unbounded series of levels, we can define the entire transfinite cumulative hierarchy for all ordinal numbers, finite and transfinite alike: D2 UA = U UgOB = Ug ∪ ℘(Ug) Uh = ⋃gihUg,	for limit ordinals γ This intuitive and deeply satisfying conception of the structure of the set theoretic universe yields a compelling explanation of Russell's Paradox: a plurality safely constitutes a set if and only if it is bounded in the full cumulative hierarchy, that is, if and only if there is a level of the hierarchy at or before which the plurality "runs out", that is, a level Ug such that, for everything x in that plurality, λ(x) is no higher than Ug - in which case those things are "available" to be collected into a set at the next level UgOB. However, some predicates - notably, 'x ∉ x' - pick out absolutely unbounded pluralities, pluralities that never "run out" by any level.19 Accordingly, as 17 L(y) is definable without any mention of finitude as: ∀a[(U ∈ a ∧ ∀b(b ∈ a → b ∪ ℘(b) ∈ a)) → y ∈ a], i.e., "y is in every set that contains the set U of urelements and also contains b ∪ ℘(b) whenever it contains b, for any set b". 18 ω is the first transfinite ordinal number, i.e., the first "counting number" after the natural numbers. It is also the first limit ordinal, i.e., the first ordinal α > 0 such that, if β < α, then β + 1 < α. See (Devlin, 1991), Ch. 3 for a good introduction to transfinite ordinals and cardinals and their arithmetic. 19 In the cumulative hierarchy, where all sets are well-founded, no set is a member of itself so 'x ∉ x' is in fact true of everything. 10 in Russell's Paradox, the assumption that they do - i.e., the assumption that they constitute sets - leads to contradiction. 3 Modal Set Theory and the Completion Problem Unfortunately, as satisfying and illuminating as this explanation might be, a serious puzzle remains for the realist who wants to take the existence of sets seriously: the full cumulative hierarchy is itself a well-defined plurality; why is it not safe? That is, why does the hierarchy itself fail to be "completed" so as to constitute a set? Note the question is not: Why is there no universal set, i.e., no set containing all the urelements and all the sets? As we've just seen, the iterative conception of set provides a cogent answer to that question: only those pluralities that "run out" by some level of the cumulative hierarchy constitute sets at the next level and, obviously, the entire hierarchy is not such a plurality; there is no level at which the members of all the levels form a set. Rather, the question is: Why is the hierarchy only as "high" as it is? Why do all the urelements and sets that there actually are fail to constitute a further level that kicks off yet another series of iterations? For the same justification we had for "completing" the hierarchy of finite levels appears to apply no Im pu re S et sImpure Sets Pure Sets ••••• • ••••• • • • • •• • • • • • •• •• • • • •• •• Uω ••••• • • U1 U0 U2 ••• ••• ••• Un Un+1 ••••• •••• Uω+1 Uω+ω Uω+ω+1 Figure 2: The transfinite cumulative hierarchy 11 less to the full hierarchy: all of the urelements and sets in all the levels are there, as robustly as the members of the finite levels; moreover, since the set they would constitute - call it Up - would be of a higher level than its members and, hence, would not be a member of itself, just as with Ua, no obvious threat of paradox looms. Call this the completion problem. Granted, the completion problem does not appear to be as grave and immediate a threat to the coherence of set theory as Russell's Paradox. But it does raise disturbing questions for the realist: if there is no answer to the completion problem, then there is an essential element of randomness to set existence. For once we acknowledge that there are pluralities that inexplicably fail to constitute sets, it is hard to see what grounds there are for picking and choosing between those that do and those that don't: in particular, if the same reasons for accepting that the finite levels UA, UB, UC, ... constitute a set seem to hold for all the levels of the hierarchy without their constituting a set, then what reason to do we have for accepting that even the finite levels UA, UB, UC, ... do? Or that any plurality does for that matter? Without a solution to the completion problem, then, the actual structure of the hierarchy appears to be unknowable; any claims to knowledge of it would appear to be groundless, as the objects of the purported knowledge might well concern entities that simply do not exist. Putnam was the first to argue explicitly that such questions are answered by taking the principles underlying the iterative conception of set to be essentially modal and, more specifically, by suggesting that a set is not to be understood in terms of the actual existence of a finished thing but as the possibility of its formation:20 [T]here is not, from a mathematical point of view, any significant difference between the assertion that there exists a set of integers satisfying an arithmetical condition and the assertion that it is possible to select integers so as to satisfy the condition. Sets, [to parody] John Stuart Mill, are permanent possibilities of selection.21 20 (Putman, 1967), p. 12. See (Hellman, 1989) for a detailed development of Putnam's ideas and (Roberts, 2018) for a rigorous critique. 21 There are intimations of Putnam's idea in Cantor, notably in an 1897 letter to Hilbert: "I say of a set that it can be thought of as finished ... if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and so to think of the set itself as a compounded thing for itself ; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements." (Ewald, 1996), p. 927. The emphasis is Cantor's. 12 Parsons (1977) spells the idea out a little less metaphorically in a thesis - call it Parsons' Principle - that addresses the completion problem directly: any given plurality of things "can constitute a set, but it is not necessary that they do." Thus, necessarily, no matter how many cumulative levels there might be, the absolutely unbounded pluralities that don't in fact constitute sets in any level nonetheless could have constituted sets. The answer to the completion problem on this potentialist conception of sets, then, is simply that there neither is nor could be a "completed" cumulative hierarchy. Rather, instead of the completed stages of the cumulative hierarchy, we have a potential hierarchy, i.e., roughly speaking, an infinite hierarchy of possibilities where, given any possible completion of the hierarchy up to a given level Ug, there is always a more expansive possibility in which some of the mere pluralities of Ug constitute sets - in the "maximal" case, a possibility comprising the next level UgOB. The completion problem only arises on the assumption that all the levels - hence all the sets - that there could be (relative to an initial set UA of urelements) are already actual and, hence, that the hierarchy of sets is complete, that there is no more "collecting" of pluralities into sets that can be done. For only under that assumption - call it actualism - is it mysterious why the hierarchy is only as high as it is, why it (or indeed any absolutely unbounded plurality) fails to constitute a further set. The potentialist rejects the actualist assumption: the unbounded pluralities of one possible world always constitute sets in further, more comprehensive worlds. What becomes of ZF on the potentialist conception? Thought of semantically, the potentialist conception suggests (roughly put) that an assertion to the effect that a certain set exists - and hence occurs at some level of the cumulative hierarchy - should be understood as the assertion that it is possible that such a set exist; likewise, assertions about all sets should be understood, not simply as assertions about the sets that in fact exist but, roughly speaking, about all the sets there could be, all the sets in any possible world.22 Formalized, this insight yields what Linnebo (2013) calls the potentialist translation φ◇ of a sentence φ of ordinary set theory, viz., the result of replacing every existential quantifier occurrence ∃ in φ with its modalized counterpart ◇∃ and every universal quantifier occurrence ∀ with □∀. The idea, then, is that, if a statement φ of ordinary ZF set theory is purportedly true in the cumulative hierarchy, its modalized counterpart φ◇ will be true in the potential hierarchy. This in turn might suggest that modal set theory will simply consist 22 The exposition from this point draws heavily upon (Linnebo, 2010; 2013) which in turn was strongly influenced by (Parsons, 1983). 13 in formulating the theory ZF◇ that results from replacing the axioms of ZF with their potentialist translations. But that would not in and of itself be terribly interesting. Instead, modal set theorists like Parsons and Linnebo opt for the far more illuminating tack of taking the potentialist conception itself as primary and, hence, by axiomatizing its fundamental principles, deriving the axioms of ZF◇. The first task toward that end is to identify the right propositional modal logic for the potentialist conception. Expressed in terms of possible worlds, the basic underlying intuition is that the universe of sets in a given world w can always be increased - for any world w there is an accessible world w′ that includes, not only everything already in w, but new sets whose members are mere pluralities in w.23 This can be captured more formally by means of several intuitive constraints on world domains D(w) and the accessibility relation R. Specifically, growth is reflected in the inclusion constraint that, if Rww′, then D(w) ⊆ D(wv).24 R itself should be a partial order (i.e., reflexive, transitive, and anti-symmetric) - each world w′ accessible from a world w represents a way in which some of the mere pluralities of w constitute fully-fledged sets in w′, but different pluralities of w might constitute sets in different accessible worlds. Moreover, that the formation of new sets proceeds discretely is reflected in the requirement that R be weakly well-founded.25 Finally, if wB and wC represent distinct expansions of the set theoretic universe of a world w - different "choices" of which mere pluralities of w to take to constitute sets - it should still be possible in each world to form the sets constituted in the other. Hence, a further natural condition is that accessibility be directed, that is, that, for any two worlds wB, wC accessible from a given world w there is a third world accessible from both and, hence, given the inclusion constraint, one whose sets include all those formed in either world. The propositional modal logic 23 For fairly obvious reasons, possible world semantics can only be considered a useful heuristic, as Kripke models are themselves by definition sets. Hence, no Kripke model could purport to represent (in a one-to-one fashion) all possible sets (over some initial collection U of urelements) without opening the door once again to Russell's Paradox. (And allowing "class-size" Kripke models would only give rise to a version of the completion problem if one claimed that such a model could include all possible sets - why isn't there a further world encompassing all the sets in that model?) This does, however, raise the further question of what it means for the potentialist translation of a theorem of ZF to be true, if it is not true in virtue of facts about genuinely existing sets in genuinely existing possible worlds. 24 This will of course have the consequence that, for any object x, including urelements, it will be a logical truth that x exists necessarily, □E!x. That is the right result, as the constraints in question are only meant intuitively to capture possible ways of extending the set theoretic universe of a given world w; nothing in w "goes away" as new sets are added in accessible worlds. 25 R is weakly well-founded if every nonempty set S of worlds contains an R-minimal world, i.e., a world w ∈ S such that, for all other u ∈ S, not-Ruw. (R is well-founded if it is a strict partial order - irreflexive and transitive - that satisfies this definition.) 14 determined by these conditions on accessibility is S4.2, the normal modal logic that includes the familiar axioms of the logic S4, viz., T □φ → φ 4 □φ → □□φ corresponding to reflexivity and transitivity, respectively, and the axiom G ◇□φ → □◇φ corresponding to directedness.26 And to S4.2 is added classical quantification theory with identity and the axiom 'x ≠ y → □x ≠ y' expressing the necessity of difference. Now, as seen above, it is useful to express the axioms of ZF informally in terms of pluralities. To characterize the potentialist conception properly with the tools of modal logic, it is essential to quantify over pluralities explicitly in order to identify the logical principles that govern their behavior. Accordingly, we introduce plural variables 'xx', 'yy', etc. and a new type of atomic formula 'y ≺ xx' to indicate that y is one of the things xx. The inference rules for plural quantifiers parallel those for first-order quantifiers exactly.27 Several principles capture the existence and nature of pluralities. First, given the ontological innocence of plural quantification, the plural counterpart to NC seems harmless:28 PC ∃xx∀y(y ≺ xx ↔ φ(y)), where 'xx' does not occur free in φ(y), that is, simply put, for any predicate φ(y), there are the things it is true of.29 Next, the modal properties of pluralities are captured in two axioms expressing the stability of the ≺ relation, that 26 There is no axiom corresponding to antisymmetry because it is not definable in propositional modal logic. See, e.g., (Blackburn, de Rijke, & Venema, 2001), §4.5. Well-foundedness can be axiomatized in a bimodal extension of propositional modal logic with both "forward-looking" and "backward-looking" necessity operators. See (Studd, 2013). 27 See (Linnebo, 2014) for a useful overview of plural quantification and (Oliver and Smiley, 2013) for a comprehensive treatment of plural logic. 28 Though see (Spencer, 2012). 29 As Linnebo points out (2013, p. 210), PC entails the existence of an "empty" plurality. This can be avoided at the cost of some inconvenience. As to PC's harmlessness, note that a Russell-style argument to a contradiction from PC with φ(y) = 'y ⊀ y' (assuming that construction is even permitted syntactically) breaks down, as it is in general invalid to infer from the fact that all things (distributively) have a property F, that any things (collectively) have F. Hence, from ∀yφ(y) we cannot legitimately infer φ(xx); in particular, we can't infer xx ≺ xx ↔ xx ⊀ xx from ∀y(y ≺ xx ↔ y ⊀ y). For more on distributive vs. collective quantification, see, e.g., (Scha, 1984) and (McKay, 2006). 15 is, that, for any plurality xx and object y in a given world w, y will be among the things xx in an arbitrary accessible world w′ if and only if it is among them in w: Stb≺O y ≺ xx → □(y ≺ xx) Stb≺ y ⊀ xx → □(y ⊀ xx). However, these axioms don't rule out the possibility that xx grows in a further world w′, that xx includes a new object z that only first comes to exist in w′. This possibility is ruled out by means of a schema that ensures that pluralities are inextensible: InEx≺ ∀y(y ≺ xx → □φ) → □∀y(y ≺ xx → φ).30 Let MPFO be this system of plural first-order modal logic. The next task is to extend MPFO to a basic modal set theory BMST. Like ZF, BMST axiomatizes the two fundamental structural properties of sets, viz., extensionality and foundation,31 which are captured simply by adopting the axioms Ext and Fnd (hence also their necessitations). Recall from Section 1 above, however, that sets also have their members essentially, as expressed in the principles E∈ and E∉.32 Both of these principles, as well as the inextensibility of membership, are entailed by the following: ED∈ ∃xx□∀y(y ≺ xx ↔ y ∈ a). Together with the stability and inextensibility principles for pluralities above, ED∈ says that, for any set a, one and the same plurality of things constitute a in every (accessible) possible world. 30 To understand how this works to ensure inextensibility it is useful to note that InEx≺ is just the Barcan formula (BF) - ∀y□φ → □∀yφ - restricted to pluralities xx. The semantic effect of unrestricted BF is that, so to say, existence is inextensible, i.e., that there couldn't have been anything that doesn't already exist in fact, a notoriously problematic metaphysical claim (see (Menzel, 2018), esp. sections 1 and 2). For suppose there could have been something a distinct from everything that exists in fact. Then ∀y□(y ≠ a). But then by BF, □∀y(y ≠ a), contradicting the assumption that such an a could have existed. Restricted as in InEx≺, however, the semantic effect is simply that, for a given plurality xx, there couldn't have been anything a among them that isn't already among them, i.e., that pluralities are inextensible. 31 Foundation follows semantically from the weak well-foundedness of the accessibility relation R and the fundamental potentialist assumption that the members of any set that first comes to be in any world w is constituted by objects that exist in some world u such that Ruw. See (Linnebo, 2013), pp. 216-17. 32 Since □E!a is a simple theorem of MPFO, the antecedents E!a of the conditionals in the consequents of these two principles are superfluous in the context of BMST. 16 Recall that the general intuitive motivation for the naïve comprehension principle NC was that the existence of the members of a purported set should suffice for the existence of the set, which we can now express generally, and formally, in terms of plural quantification: GNC ∀xx∃a∀y(y ∈ a ↔ y ≺ xx). Given the innocent principle PC, NC (and catastrophe) follow immediately from GNC. As we've seen, both actualist and potentialist accounts have explanations of where GNC goes wrong based intuitively on the iterative conception of set. On both accounts, in one sense or another, sets are "constructed" level-by-level without end, the new sets of one level constituted by the mere pluralities of previous levels. Hence, necessarily, the plurality of all the sets in all the levels, and unbounded pluralities generally, fail to constitute sets. Both accounts thus agree on where GNC gets it wrong: necessarily, there are pluralities that don't constitute sets. Additionally, however, the potentialist conception avoids the completion problem by showing that GNC almost gets it right: it's not that every plurality does constitute a set; it's that every plurality could. From the potentialist standpoint, then, GNC simply missed the implicit modality in claims of set existence; what we need is just its potentialist translation: C □∀xx◇∃a□∀y(y ∈ a ↔ y ≺ xx). That is, necessarily, any plurality of things could constitute a set consisting (necessarily) of exactly those things. This, of course, is a precise formalization of Parsons' Principle. Let BMST be the result of adding the axioms Ext, Fnd, ED∈, and C to MPFO. It is a simple matter to show that BMST proves the potentialist translations of all of the ZF axioms except Pow, Inf, and Rep.33 As these are by far the most powerful axioms of ZF, this shows that our basic intuitions about pluralities and sets as expressed in BMST - in particular, Parsons' Principle C - only get us so far. To see this with regard to Pow, suppose we have a plurality xx in some world w. By principle C there is a world u accessible from w in which they constitute a set b. Obviously, all the "subpluralities" of xx - all the pluralities yy such that each thing in them is among the things xx - also exist in w. But neither C nor the structure of worlds and their domains in the underlying semantics provides any guarantee that any of them (other than 33 See (Linnebo 2013, p. 221) and (Parsons 1983), Appendix 2. The addition of the urelement axiom Ur, which neither Linnebo nor Parsons includes in their discussions, would involve no significant complications. 17 xx) constitute a set in u, let alone all of them. Indeed, for all we know on the potentialist conception to this point, the subpluralities of xx might be inexhaustible in the sense that, for any world u accessible from w, there is always a further world v accessible from u in which some subplurality of xx only first constitutes a set (hence, a subset of b). If so, there is no world where all possible subsets of b exist, in which case the power set of b is impossible, contrary to Pow◇, i.e., ◇∃a□∀x(x ∈ a ↔ x ⊆ b). To derive Pow◇, then, an additional principle is required that rules out this sort inexhaustibility. Intuitively, this is accomplished most naturally by assuming that worlds more directly reflect the levels of the cumulative hierarchy; that is, by assuming, not only that the newlyformed sets of a given world are mere pluralities of some preceding world, but that set formation is always maximal: that the newly-formed sets of a world are all those that can be constituted from the mere pluralities of a preceding world.34 This assumption can be expressed elegantly in a single axiom to the effect that, much like its members, the subsets of a given set a are constant across possible worlds: ED⊆ ∃xx□∀y(y ≺ xx ↔ y ⊆ a). Pow◇ now follows straightaway from BMST + ED⊆. Recall that ES and Pr alone suffice to generate the infinite plurality of Zermelo numbers ∅A, ∅B, ∅C, ... and it was left to Inf simply to sanction a set containing them. If we could prove the mere possibility that all the Zermelo numbers exist on the potentialist conception, we could immediately invoke principle C to prove the possible existence of a set containing them. However, the potentialist is in a slightly more fraught situation. For the potentialist principles to this point - the derived principles ES◇ and Pr◇ in particular - only yield a series of possible initial segments of the Zermelo numbers: by ES◇ it is possible that ∅B exists; and by iterated applications of Pr◇, it is possible that ∅A and ∅B exist and hence also that the numbers ∅A, ∅B, and ∅C exist, and so on, but without the entire series of Zermelo numbers ever being "completed" in a single possibility. 34 Linnebo (2013, p. 209, fn 7) suggests that maximality entails that the accessibility relation is linear but that would follow only if worlds are individuated extensionally by their domains (which is plausible if, as in (Linnebo 2013), the existence of urelements is not assumed). Linearity can be forced axiomatically by any of a variety of axioms; see (Chellas, 1980). The system resulting from adding a linearity axiom to S4 in place of G is S4.3 though, as Parsons notes (1983, pp. 319-20), S4.3 seems to add nothing over S4.2 for the purpose of deriving set-theoretic consequences. 18 To derive the possible existence of all the Zermelo numbers thus requires a further principle asserting, roughly, that whatever is true of the potential hierarchy as a whole, as expressed in the potentialist translation φ◇ of some proposition φ of set theory (hence a proposition containing no plural quantifiers35), is possible simpliciter: Ref φ◇ → ◇φ. Thus, in particular, ES◇ and Pr◇ yield a proposition φ◇ expressing the infinite series of possibilities involving larger and larger initial finite segments of the Zermelo numbers.36 By Ref that series is reflected in a single possibility containing all - hence, by PC, the entire infinite plurality of - the Zermelo numbers, and so by Parsons' Principle C it is possible that they constitute a single infinite set. Inf◇ follows immediately by some simple modal logic.37 The potentialist translation Rep◇ of the Replacement schema can be similarly proved by strengthening Ref to RefO ◇∀x(φ◇(x) → φ(x)).38 Importantly, it can be shown that the modalized quantifiers ◇∃ and □∀ "behave prooftheoretically very much like ordinary quantifiers" (Linnebo, 2013, p. 213), thus explaining why they are not found in ordinary set-theoretic practice - mathematicians can, in effect, talk about the potentialist hierarchy as if it were actual. Concluding Philosophical Postscript. This article has focused chiefly on the technical development of modal set theory and its intuitive motivations without any close critical attention paid to surrounding philosophical questions. In closing we note briefly that a problem parallel to the completion problem threatens to arise for the potentialist. To see this, first, instead of restricting ourselves, relative to any given possible world, only to those possibilities that represent growth of the set theoretic universe, as we do in characterizing the potentialist hierarchy, let us broaden our 35 Allowing plural quantifiers in φ without restriction would in fact lead to Russell's Paradox. 36 Specifically, φ◇ is ◇∃a∀xx ∉ a ∧ □∀x◇∃y□∀z(z ∈ y ↔ z = x) and hence ◇φ is ◇(∃a∀x x ∉ a ∧ ∀x∃y∀z(z ∈ y ↔ z = x)). 37 See (Wigglesworth, 2018) for an account that, unlike those of Parsons, Linnebo, and Studd, doesn't require any sort of reflection principle to prove the possibility of an infinite set. 38 Again, where φ contains no plural quantifiers. Parsons (1983, p. 323) relies on a rather different principle to derive Inf◇ and Rep◇. See (Linnebo 2013), p. 223, fn 28 for important further discussion. 19 metaphysical perspective to one where we are considering all possibilities on a par. From this standpoint, we see that, on the potentialist conception, at least the vast majority of pure sets are metaphysically contingent beings - in particular, for any given possible world w, those pure sets in any accessible world w′ whose members are mere pluralities in w. If this is in fact the sober truth about the metaphysics of sets, then set existence is metaphysically capricious - two possible worlds can be in all respects identical but for the fact that there are pure sets in one that simply and inexplicably fail to exist in the other. Though not identical to the completion problem, the apparently inexplicable contingency of set existence on the potentialist conception, taken literally, seems to raise questions parallel to those arising from the apparently inexplicably nonexistence of certain sets, as noted in the completion problem. Perhaps in response to this difficulty - though neither explicitly says so - both Parsons (1977, §IV) and Linnebo (2013, pp. 207-8) suggest that the modality of the potential hierarchy is more semantic than metaphysical: at any given time, one's conception of the "height" of the set theoretic universe, hence the range of one's quantifiers, is determined by one's strongest large cardinal assumptions.39 Once convinced of the existence of a larger cardinal still, pluralities that had been (relative to the earlier conception) absolutely unbounded constitute sets under the stronger assumptions and the range of one's quantifiers broadens accordingly. Thus Linnebo (p. 208): A claim is possible, in this sense, if it can be made to hold by a permissible extension of the mathematical ontology; and it is necessary if it holds under any permissible such extension. Metaphysical modality would be unsuitable for our present purposes because pure sets are taken to exist of metaphysical necessity if at all. However, if after all (pure) sets exist as a matter of metaphysical necessity (so that, in particular, any pure sets that could have existed actually exist), as Linnebo appears to suggest here, then the completion problem threatens once again to rear its head with all its original force: why are there only the pure sets there actually are? If, necessarily, all pure sets exist of metaphysical necessity, 39 See also (Fine, 2006). 20 what explains the fact that there couldn't have been more, the fact that there are pluralities of things such that it is not even metaphysically possible that they constitute sets?40 These and related metaphysical questions prompted by the potentialist conception of set point to a fertile area for exploration in the philosophical foundations modal set theory. References Blackburn, P., de Rijke, M., & Venema, Y. (2001). 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Volume-I, Issue-I July 2014 25 International Journal of Humanities & Social Science Studies (IJHSSS) A Peer-Reviewed Bi-monthly Bi-lingual Journal of Humanities & Social Science ISSN: 2349-6959 (Online), ISSN: 2349-6711 (Print) Volume-I, Issue-I, July 2014 Published by Scholar Publications, Karimganj, Assam, India, 788711 Website: http://www.ijhsss.com Matrimonial Alliances between the royal houses of Tripura and Manipur in the days of monarchy Memchaton Singha Assistant Professor Dept. of History Rabindrasadan Girls' College, Karimganj Abstract Manipur and Tripura are two important integral states of North east India with a long glorious history of its own. Both the states maintained cordial relationship from the very early period despite regular intervals of conflicts. Marriage alliances between the royal families of Manipur and Tripura were common phenomena which began from remote antiquity and continued till 20 th century. The instance of first marriage alliance occurred during the reign of Tripura king Taidakshin, the 43 rd Raja. However, the most notable cases of matrimonial alliances between Manipur and Tripura started towards the end of 18 th century. The distinctive feature in the marriages was that the Rajas of Tripura married not only Manipuri princesses but also many Manipuri girls belonging to plebeian family. The marriage alliances not only brought cordial relations between the two kingdoms but also there was cultural assimilation. The tradition of marrying Manipuri girls were even followed by many noblemen of the Tripura royal family. Tripura also witnessed various public welfare works that was contributed by the Manipuri Maharanis. Thus, we find that the Tripura royal house was filled with descendants from the Manipuri queens. This article focuses on providing a chronological historical account of the matrimonial alliances that occurred in the royal families of Manipur and Tripura during the period of monarchies. It also highlights the basic causes behind the arrangement of the marriages and also the cultural amalgamation between the two states. Keywords: Royal families, matrimonial alliances, monarchies, wed-lock, Meitei Queens, inter-dynastic. Tripura is one of the important seven sisters‟ states of North East India. The state in bounded on the north, west, south and south-east by Bangladesh. In the east Tripura shares boundary with the state of Assam and Mizoram. About the state of Tripura we find mention in the famous epic the „Mahabharata‟. This naturally highlight about a very ancient origin of the kingdom. Tripura was ruled by various commanding rulers who thrived in setting a landmark in the historical arena of India. According to the royal chronicle of Tripura "Sri Rajmala" and other ancient literatures of the state the dynasty trace their origin from the Lunar dynasty and from the great king Yayati. However, Druhyu, the son of king Yayati laid the foundation of Tripura dynasty. The kings of Tripura began to suffix the title of "Manikya" from the reign of Ratna Fa or Ratna Manikya. The descendents of Ratna Manikya ruled the kingdom without much interference from the British. But, in 1765 A.D. the opportunity aroused for the British and Tripura was brought under the British colonial rule. However, it was not kept under the direct control by the British authorities. The state existed as an independent princely state and maintained its integrity unless the kingdom joined in the Indian Union on Oct. 15, 1949. Matrimonial Alliances between the royal houses of Tripura.............. Memchaton Singha Volume-I, Issue-I July 2014 26 Manipur like Tripura is an important state of North East India. Pandit Jawaharlal Nehru has called the land as „Jewel of India‟ as well as a melting pot of rich culture. Manipur is bounded by Nagaland in the North, Mizoram in the South, Cachar district of Assam in the West and Burma in the East. The state has long glorious history of its own. The land is inhabited by many tribes with different cultural identity and ethnicity. The kingdom of Manipur was developed after a struggle of conflicts between various ethnic tribes and clans or Salais in the valley. The numerous groups were later amalgamated and brought under the political sovereignty of seven leading clans namely Ningthouja (Mangang), Luwang, Angom, Khuman, Moirang, Kha Ngamba and Sarang Leisangthem or (Chenglei). These clans were powerful at some point of time but by the first half of 15 th century the Ningthoujas or Meiteis succeeded to overpower all the other clans. Thus, with this the other clans or social groups were brought within the social structure of this dynasty under the common ethnonym „Meitei‟. The first recorded king of Manipur was Nongda Lairen Pakhangba who is said to have ruled the kingdom from 33 A.D. The history of Manipur reached its glory to its zenith in the field of military, religion, culture and literature during the reign of Pamheiba alias Garib Niwaz (1709-1748 A.D.). Hinduism was adopted as state religion during his reign. The state has maintained the status of princely independent state until it became a British protectorate princely state only after the battle of AngloManipuri war or the „Khongjom war‟ of 1891 A.D. Finally, it was in 1949, after two years of Indian independence that Manipur was merged to Republic of India. The states of Manipur and Tripura thus formed two most important states of pre-colonial North East India. From antiquity both the states is said to have maintained friendly relations. During their princely regime they were neighbouring states and came into contact with each other in many occasions such as for trade, diplomacy, dwelling, matrimony etc. Moreover, an assortment of crucial political circumstances demanded the kingdoms to come in close contact with each other for political interdependence. Such as, when there were rapid incursions of Burmese in Manipur and that of Pathans and Mughals in Tripura both the kingdoms depended on each other for relief from the intruders. The political interdependence between the two states was boost up by the policy of the marriage alliances. The tradition of wed-lock between the two kingdoms began since antiquity from the time of 43 rd Tripura Raja Taidakshin and the tradition continued till 20 th century. In the early period Manipur was called by the Tripuris as Mokhali/Mekhali and to the Manipuris the kingdom of Tripura was known as Takhen/ Takhel. The word „Takhen‟ has come from the Bengali word „Dakhin‟ which means South. The people of Sylhet used to call Tripura „Dakhin‟ which was pronounced by the Manipuris as Takhen. They deviate the„d‟ sound changing to‟t‟ and that of 'i' sound changing to 'e' due to their speech habit. However, Naorem Sanajaoba, an eminent scholar of Manipur opined that perhaps, the Manipuris pronounced the unfamiliar name „Taidakhin‟ as „Takhel‟ or „Takhen'. And in the course of time, Takhen and Takhel came to stand for Tripura. The relation between the two kingdoms was filled with mutual co-operation, occasional conflicts and war fares. There were several cases of hostilities between Tripura and Manipur particularly between 16 th to 18 th centuries, they were in 1533, 1634, 1696 and 1724 A.D. The most remarkable armed clash between the two took place during the reign of Tripura king Dharma Manikya II (1717-1733 A.D.) and the Manipuri king Garib Niwaz or Pamheiba (1709-1748 A.D.). In this battle Tripuris suffered severe defeat in the hands of the Manipuri troops. In order to commemorate the glorious victory over Tripura, a book was written called "Takhel Ngamba" (Conqueror of Tripura). However, despite such hostilities, the traditions of wedlock between the two kingdoms were common phenomena since centuries back. In the light of history the first reference of matrimonial alliances between the royal houses of Manipur and Tripura is recorded in the royal chronicle of Tripura "Sri Rajmala" that Tripura Raja Matrimonial Alliances between the royal houses of Tripura.............. Memchaton Singha Volume-I, Issue-I July 2014 27 Taidakshin, grandson of king Trilochana married a princess of Mekhali (modern Manipur). But, it is unfortunate that there is absence of supportive sources about the event of the marriage. Again, due to divergence in the chronology of Tripura kings an authentic date cannot be fixed. After this we do not find references of matrimonial alliances between the two kingdoms for many centuries. Scores of years later the state chronicle of Manipur "Cheitharol Kumbaba" recorded for the first time that in 1609 A.D. Akhoicham girl „Yangnu‟ was married to king of Takhen (modern Tripura). It can be contemplated as per the date of the marriage that the ruling monarchs of the kingdoms were perhaps Meitei king Khagemba (1597–1652 A.D) and the Tripura Raja Yosodhar Manikya (1600-1623 A.D.). In the following years there were many references about diplomatic correspondences between the two states but it was until 18 th century that we find rare mention of any interdynastic marriage alliances. But, Nalini Ranjan Roychoudhury has mentioned about sending an envoy in 1662-63 A.D. to the court of Manipur accompanied by one girl and one elephant. But no further information is provided on the issue. However, the most prominent cases of matrimonial alliances between Manipur and Tripura started towards the end of 18 th century. At the end of the 18 th century in 1785 A.D., Rajdhar Manikya II ascended the throne of Tripura and ruled up to 1804 A.D. In Manipur, his contemporary Raja was king Bhagyachandra alias Joy Singha (1759-1761 & 1763-1798 A.D.). During their rule the first historically significant marriage union took place. The old Raja Bhagyachandra of Manipur was a religious individual and after installing his eldest son Prince Labanyachandra as new king of Manipur started for pilgrimages to numerous secret holy places of India in 1798 A.D. On his journey he also visited Tripura and halted back at Agartala, the capital for some days. During his stay he gave one of his daughter princess Hariseswari in marriage to Tripura king Rajdhar Manikya II. Thus, with this the old tradition of wed-lock between the royal houses was renewed. The „Kanya Dana‟ or „Girl offering ceremony‟ of Princess Hariseswari was performed by her brother Sri Tulajit. This marriage was effective in bringing genial relations among the states. Following the wedding, Raja Joy Singha did not extend his stay for long at Agartala and started for his pilgrimage. The journey thus being through water ways the Tripura Raja helped him by supplying 15 boats for the river journey and also gave seven hundred silver coins to Raja Bhagyachandra for his travelling expenses. After the marriage alliance road connecting Tripura and Manipur were widened and easy immigration between the states became constant. It was in 1804 that the Meiteis even helped the Tripuris in suppressing raids made by the Kukis in the Tripura border. The new Queen Hariseswari brought all along the idol of "Sri Radhamadhav" from her parental house which was installed in the royal compound. The king brought the priest of the temple, some musicians, some noble men and maid attendants of the Queen from Manipur and gave them settlement near the royal house at a village called „Mekhlipara‟ (village of Manipuris). But, unfortunately, at present no Mekhli/Meitei inhabits in the village. When the capital of the kingdom shifted from old Agartala to the new Agartala (Nutan Haveli) during the reign of king Krishna Kishore Manikya (1829-1849 A.D.) the deity of the "Sri Radhamadhav" was brought to the new place and installed to the north of the royal palace. After the name of the Radhamadhav the place came to known as „Radhanagar‟. Many Hindu festivals like „Maharasa‟ (of various forms) are performed in this temple every year like an imitation as done in „Sri Sri Govindajee‟ temple of Manipur. After the death of Rajdhar Manikya his son Ram Ganga Manikya succeeded the throne of Tripura. The king did not marry any of the Meitei girls like his father. The footstep of marrying Manipuri girls by the Tripuri Rajas continued thereafter. The distinguished verity of the later marriages was that the Tripura Rajas not only married the princesses of the royal family of Manipur but also the Meitei maidens who have migrated and settled in Tripura due to political instability in their native or for economic necessity. Matrimonial Alliances between the royal houses of Tripura.............. Memchaton Singha Volume-I, Issue-I July 2014 28 The major Meitei migration in the kingdom of Tripura took during the Burmese invasion of Manipur in 1819 to 1825 A.D. which is called in the history of Manipur as the „Seven Years Devastation‟, many Manipuris had to flee from their native land and took settlement in the nearby kingdoms. Among them numerous obtained shelter at Tripura and later became permanent domicile of the land. The then Raja of Tripura Kashichandra Manikya (18261829 A.D.) was fascinated by the beauty, cleanliness and plethora of health of the Manipuri girls. In 1826 A.D., being captivated by the beauty of Meitei princess Kutilakha/Kutilakshi of Manipur, the king married her. The king also married three other Manipuri girls who were from plebeian families. The Meitei Queen Kutilakshi gave birth to a son named Krishnachandra who was appointed as Barthakur in later years. But it was unfortunate that the prince Krishnachandra died at his early age. Kashichandra Manikya also had many other wives from different communities. Prince Kashichandra Manikya also married the daughter of the king of Assam named Ratnamala on 21 st December 1822. The retinue of the princess was provided settlement near „Mekhlipara‟ and was called Assam-para and to the Manipuris, it was known as Tekhao Leikai (village of Assamese). After the death of Raja Kashichandra Manikya, his son Krishna Kishore Manikya (18291849 A.D.) ascended the throne of Tripura. Similar to his forerunners, he also married many Manipuri girls and increased the number of Meitei Queens in the royal house of Tripura. Of the Queens three of them were the daughters of Manipuri king Marjit Singha, they were – princess Chandrakola, princess Vidhukola and princess Akhileswari. Queen Akhileswari gave birth to prince Nilkrishna. The three princesses were perhaps settling in Sylhet along with their father prior to commencement of the marriages. Maharaja Krishna Kishore Manikya at his old age also married a beautiful Manipuri Brahmin girl called Purnakala who was a cook in the Tripura royal kitchen. The Manipuris called the new Rani as „Bamon Leima‟ or „Brahmin Queen‟. This marriage became a matter of discussion and criticism among the subjects in the kingdom. Curved with numerous problems Raja Krishna Kishore Manikya did not take the new Maharani to the royal fortress. Therefore, he built a house at Haidra (Dholeswar) for the Rani where she was kept along with some attendants and sepoys. The place „Haidra‟ was situated 5 miles in west of modern Agartala, which was a forest area full of woods and birds and was a convenient place for hunting and games. Krishna Kishore Manikya being a lover of hunting visited the place severally. Atlast the king being fascinated and attracted towards the beauty of the place decided to shift his capital there and the construction of the new capital started in 1838 A.D. Thus, present Agartala town had its origin in the growth of Manipuri community. Again, it is very obvious that Rani Purnakala became one of the factors behind the establishment of the new capital. One notable event was that one Kumari Urmilla, daughter of Krishna Kishore Manikya was married to Kumar Tilok, son of Duljit (Daoji) who was one of the grandsons of Maharaja Bhagyachandra. In the realm of Tripura-Manipuri matrimonial alliances it was the first instance of marrying Tripuri girls by the royal descendants of the Meities. However, in future years we do not find any such references of reciprocal marriage. Ishan Chandra Manikya son of Krishna Kishore Manikya ruled Tripura from 1849 to 1862 A.D. The new king also followed the long tradition of wed-lock with the Manipuri girls. He had four Maharanis one of whom was Bengali lady named Rajlakshmi, the other three were all Manipuri khsatriyas viz. Moirangthem Chanu Muktabali, Keisam Chanu Jatiswari and Khumanthem Chanu Chandreswari. Maharaja Ishan Chandra Manikya did not have any child from his first wife Rajlakshmi. His second wife Muktabali Devi gave birth to prince Brajendrachandra. From his fourth wife Jatiswari prince Nabadwipchandra and from his third wife Chandreswari prince Rohinichandra was born. Maharani Jatiswari is mentioned to have 3 more girl child along with Nabadwipchandra. Maharani Muktabali was Matrimonial Alliances between the royal houses of Tripura.............. Memchaton Singha Volume-I, Issue-I July 2014 29 a girl of Moirngthem clan who inhabited at modern Srimangal area of Bangladesh. The famous Manipuri musician of the time Babu Moirangthem Babuni was her nephew. He was one of the teachers who introduced „Rasa‟ dance in Tripura. Maharani Chandreswari, another queen of Maharaj Ishan Chandra Manikya was the daughter of Khumanthen Khirod who was a settler of Kasba (Keilashgar, Tripura). After the death of Maharaja Ishan Chandra Manikya his brother Birchandra Manikya (1862-1896 A.D.), the next king did not stay aloof from totaling Manipuri women in the regal fortress of Tripura. He is also considered as the maker of modern Tripura. The new king not only continued the tradition of marrying Manipuri girls but also encouraged the development of Manipuri culture in Tripura as the king being a patron of art, literature and music. Birchandra Manikya had many wives, of which there were three Manipuri Maharanis and one Rani. The Meitei Maharanis were Ningthem Chanu Bhanumati, Panganbam Chanu Rajeswari or Kaboklei and Khuman Chanu Manmohini. Maharani Bhanumati was most favorite Queen of Maharaja Birchandra Manikya. She was the daughter of Rajkumar Ningthem Kulendra of Kasba (Keilashgar) and sister of Rajkumar Ranadhwaj who was one of the influential ministers of Birchandra Manikya. Ningthem Kulendra was the son of Rajkumar Tilok who married Tripuri princess Urmila. Maharani Bhanumati gave birth to prince Samarendra who was appointed as Barthakur. It is said that Maharani Bhanumati had a premature death in 1882 A.D. which stricken the king with great grief. The depression and cry in the heart of raja made him to compose many narratives which was compiled and came out in the form of novel called "Prem Marichika Kabya". At that juncture of time the young poet Rabindranath Tagore presented the raja an anthology called "Bhangnahriday" (broken heart). Panganbam chanu Rajeswari alias Kaboklei was the second Maharani of the king. She was also known as „Dityo Ishwari‟. The father of Maharani Rajeswari was Panganbam Premsingha Thakur, a settler of Sylhet. Maharani Rajeswari gave birth to three sons, they were – Kumar Radha Kishore, Kumar Debendra and Kumar Nripendra. The eldest son of the Queen prince Radha Kishore succeeded the throne of Tripura after his father. The area which was at the disposal of the Queen was named as „Naran Rajeswaripur‟. The third and the youngest Queen of raja Birchandra Manikya was Khuman Chanu Manmohini alias Khuman Chanu Tharo. She was the daughter of Khumanthem Kirtidhvaja and was one of the sisters of Maharani Bhanumati. Maharani Manmohini was only 13 years and Maharaja Birchandra Manikya was at the age of 50 when the marriage was solemnized. The king gave the land Banamalipur as her share. She also established a temple and a „mandapa‟ near the present Iskon temple at Tripura. Birchandra Manikya died in 1896 A.D. After nine years of the death of her husband Manmohini also breathe her last and her shraddha ceremony was performed by her 14 years old son prince Brajanidhu. During the reign of Birchandra Manikya many development work of Manipuri culture was promoted. He encouraged his Manipuri Maharanis to construct temples and „mandapas‟. Maharani Rajeswari installed the „Sri Sri Radhamadhab‟ temple at Dholeswar and „Lainingthou Pakhangba‟ temple at Banamalipur. The temple is known to the nonManipuris of Tripura as „Pagla Debota‟ or „Mad God‟. It is also noteworthy that the Manipuri traditional festival „Lai Haraoba‟ was introduced for the first time in Tripura by Maharani Rajeswari. The third Queen of Birchandra Manikya, Maharani Khuman Leima Manmohini also installed Sri Sri Radha Gobinda at Banamalipur. After the death of Birchandra Manikya on 11 th December, 1896 A.D. he was succeeded by his son Radha Kishore Manikya (1896-1909 A.D.) whose mother was Maharani Rajeswari. Radha Kishore like his antecedents married many Manipuri girls. He had three Manipuri Maharanis and six Manipuri Ranis (total-9). The three Maharanis were – Ningthem Chanu Ratna Manjuri or „Dhaka Rani‟, Thongam Chanu Tulsibati and Thongam Matrimonial Alliances between the royal houses of Tripura.............. Memchaton Singha Volume-I, Issue-I July 2014 30 Chanu Manmanjuri of Nalgaria. However, the names of the other six Meitei Ranis are not available anywhere. Maharani Ratna Manjuri was called by the Meiteis of Tripura as „Dhaka Rani‟ which means „Queen from Dhaka‟. She was the daughter of the exiled king of Manipur Maharaja Debendra Singha. He ruled the kingdom of Manipur from 1849 to 1850 A.D. The Tripura king Radha Kishore Manikya married the daughter of the exiled Meitei king ahead of his succession to the throne. During the lifespan of his father Birchandra Manikya, in 1875 A.D., the king arrived at Dhaka to meet Lord Northbrook the then Governor General to discuss on various issues. During this trip, Birchandra Mankikya also visited the palace of exiled king Debendra to congregate with the Meiteis settled there. Birchandra Manikya saw princess Ratna Manjuri, the beautiful 10 year daughter of the dethroned king. He was charmed by the beauty of the princess and decided to make her his daughter-in-law. Thereafter, he brought the princess at Agartala and arranged her marriage with his son Maharajkumar Radha Kishore. Thus, Ratna Manjuri became the first Manipuri Queen of Radha Kishore Manikya. The new Queen was accompanied by her brothers namely Damanjit Rajkumar, Chitragupta (Amusna) etc. and the king arranged their settlement at Keilashgar. The area near Bishalgarh where the relatives of Ratna Manjuri settled is at present called as „Dhakarbari‟. Maharani Ratna Manjuri was the mother of the next Tripura king Birendra Kishore Manikya. The Queen built the temple of „Lamdem Lairembi‟ at Bajalghat. In 1916 A.D. at age of 75 the Queen died at Agartala and the shraddha ceremony of the deceased Queen was performed under the initiative of her grandson Maharaja Bir Bikram. The most important and revolutionary Queen of Radha Kishore Manikya was Maharani Tulsibati. Unlike other, Tulsibati was a girl of simple cultivator family of village Tarou (Nalgoriya) situated a little far away from Agartala. Her father was Thongam Rupananda who inhabited along with his wife and five daughters at the village. After the immediate death of Rupananda the responsibility of the family fell upon Tulsibati and her mother. But, soon the misfortune was swept away and Maharaja Radha Kishore Manikya married the cultivator girl Tulsibati. The Queen made remarkable contributions in the educational history of Tripura. She was a great social activist and kindhearted person. Even today she is remembered by the people of Tripura for her various contributions for the development of the society. She is credited for initiating the spread of women education in Tripura. During the last part of 19 th century a wave for the development of women education in Tripura was initiated by the Maharani. With her effort she succeeded in establishing the first girls‟ school in Tripura called "Maharani Tulsibati Balika Vidyalaya" in 1894 A.D. at Agartala. Added, Maharaja Radha Kishore Manikya also established vocational training schools where handloom training was given priority. In 1905, the Maharani established a separate woman cell at the Victoria Memorial Hospital (presently known as I.G.M) at Agartala. Maharani Tulsibati also established a market and to commemorate her contribution the name of the bazaar was kept as "Ranirbazar" (Market of the Queen). Thought Maharani Tulsibati was an uneducated person but she was expert in compositions of songs and poems. Her composed songs related to „Holi‟ festival were very famous during the time. The temple of „Sri Sri Radhamadhad‟ at Nalgaria and „Lainingthou Puthiba‟ at Abhoynagar was installed by the Queen. Beside these, we find that in almost all the Manipuri villages of Tripura such as Bishalgarh, Bamutia, Murabari, Daccabari etc. the Manipuri Queens erected a number of temples and mandapas. The Manipuri Queens brought along with them the culture and religion of their parental state which they enriches even in the royal palace of Tripura. The next Tripura Raja Birendra Kishore Manikya (1909-1923 A.D) had six Maharanis who belonged to the Nepali community and ten (10) Manipuri Ranis. Though the king married Manipuri girls like his predecessors but he did not confer them the status of „Maharanis‟. The year 1909 A.D. was a memorable and historical moment for both Tripura Matrimonial Alliances between the royal houses of Tripura.............. Memchaton Singha Volume-I, Issue-I July 2014 31 and Manipur. The then Manipuri king Churachand Singh paid a visit to Tripura on the occasion of coronation ceremony of the new Tripura king Birendra Kishore Manikya. Since years back the Manipur royal house with that of the Tipperah family had been related by marriage. It was the first occasion that the Heads of both the families met one another as relatives. The Manipuri king also visited many Manipuri villages of Tripura and rendered financial help to the settlers. Maharaja Birendra Kishore Manikya died in 1923 A.D. and the young prince Bir Bikram Kishore Manikya succeeded his father on 14 th August 1923 A.D. Bir Bikram Kishore had seven (7) wives (2 Maharanis and 5 Ranis). Among the five Ranis, three of them were Manipuris. Beside the Rajas many nobilities of the Tripura royal family also married Manipuri girls. Some of them were Maharajkumar Navadipbahadur, Brajendra Kishore alias Lalukarta, Ramendra Kishore alias Nimukarta who married Nirupama Devi, Bhadravati Devi and Jotirani Devi respectively. Maharajkumar Navadipbahadur was the son of the Manipuri Queen Jatiswari, the third wife of Ishan Chandra Manikya. Nirupama Devi was mother of famous Indian musician Sachin Deb Barman (S.D. Barman). Thus, we find that there were many cases of Manipuri girls who were taken as wives by the Tripuri Rajas and its nobilities. But, it is worth mentioning that no Manipuri king married Tripuri princess except the case of Rajkumar Tilak kumar Singha as mentioned above. Though it was alike one way in the inter-dynastic marriages between the two kingdoms but they maintained mutual and friendly relations. When Maharaja Churachand of Manipur died, many Meitei nobles from Tripura were sent to attend the shraddha ceremony and vice-versa on the death of Bir Bikram Kishore Manikya, the then Meitei king Bodhachandra visited Tripura and participated in the shraddha ceremony in 1947 A.D. M.K Binodini (daughter of Maharaja Churachand) in her introductory note for the book "Itihaser Aaloke Tripura-Manipur, Itihaski Mityengdagi Tripura-Manipur" by L.Biramangal Singha and Pannalal Roy has revealed that her grand-mother (in relation) Dhakarani alias Ratna Manjuri made her visit to Manipur. The Tripuri Queen being a representative of both the states made a marriage proposal for Tomal Manjuri (elder sister of M.K Binodini) with her grand-son Maharaja Bir Bikram Kishore Manikya. The then Raja of Manipur Churachand accepted the marriage proposal to hand over his daughter. But, unfortunately the marriage could not be solemnized. The sequence of Tripura Rajas of marrying Manipuri girls not only has political significance but also cultural. Meitei culture was promoted and extended in whole the kingdom to great extent. They introduced several Vaishnava festivals like Holi, Rasa Dance, Rathyatra, Jhulan etc. During the reign of Radha Kishore Manikya Manipuri „Vasanta Ras‟ dance was shown to Rabindranath Tagore in his honour. The poet was mesmerized by beauty of the dance. The Manipuri Dance form was immediately introduced at the initiative of the poet in the Institution of „Santiniketan‟. Thus, the historical evidences revealed that there were plenty of cases where matrimonial alliance occurred between the royal families of Manipur and Tripura during the rule of the monarchs. The Tripura royal house was filled with many descendents from the Manipuri Queens. The unique character of some of the marriages solemnized was that the Manipuri ladies whom the Rajas married simply belonged to plebeian families. Such instances reveal that the Tripuri Rajas were desperately charmed and fascinated by the beauty of the Meitei girls. The elegant ethnicity and tradition of the Meiteis also engrossed the members of the royal family and interested them to develop such wed-lock. But, it is also a fact that all the Manipuri ladies were not always bestowed the rank of the Chief Queen. Despite the detailing of the marriages occurred between the two kingdoms it is unfortunate that the unavailability of proper related materials regarding most of the marriages organized kept a part of history vacuum. Matrimonial Alliances between the royal houses of Tripura.............. Memchaton Singha Volume-I, Issue-I July 2014 32 References: Adhikari, Udainarayan., "Social – Cultural Relations Among States in Pre – Independence India, A Study of Tripura & Manipur", Akansha Publishing House, New Delhi, 2010. Bhattacharyya, A.C., "Progressive Tripura", Tribal Research and Cultural Institute, Govt. of Tripura, 2012. Bihari, Nepram., "The Cheitharol Kumbaba, The Royal Chronicle of Manipur", Spectrum publications, Guwahati, 2012. Dun, Captain, E.W., "Gazetteer of Manipur", Vivek Publishing House, Delhi, 1981. Goswami, Dr. Dwijendra Narayan., "Rajarshi Bhagya Chandra", Akkhar Publication, Agartala, 2002. Goswami, Dr. D.N. & Dr. Debbarman., "Tripura State Administration Report (1904-05, 1906-07, 1907-08)", Tribal Research and Cultural Institute, Govt. of Tripura, 2007. Hamilton, Francis., "An Account of Assam: With Some Notices Concerning the Neighbouring Territories. First Compiled in 1807-1814", Government of Assam in the Department of Historical and Antiquarian Studies, Narayani Handiqui Historical Institute, 1940. Khelchandra, N., & Ibungohal, L., "Cheitharol Kumbaba", Sahitya Parishad, Imphal, 1989. Kabui, Gangmumei., "History of Manipur, vol-I, Pre-Colonial Period", National Publishing House, New Delhi, (3 rd edi.), 2011. Menon, K.D., "Tripura District Gazetteers", Govt. Of Tripura". Agartala, 1975. Nath, Dr. N.C., "Sri Rajmala", Vol-I to IV, (Translated), Tribal Research Institute, Govt. of Tripura, Agartala, 1999. Roychoudhury, Nalini Ranjan., "Tripura Through the Ages, A short History of Tripura from the earliest times to 1947 A.D.", Sterling Publishers Pvt. Ltd., New Delhi, 1983. Roy, Jyotirmoy., "History of Manipur", Firma KLM private Ltd., Calcutta, 1999. Sen, Sri Kaliprasanna., (edited) "Sri Rajmala", Vol-I, II, III, IV, Upajati Sanskriti Gobeshana Kendra, Govt. of Tripura, 2003. Singha, Sri Kailashchandra., "Rajmala Ba Tripurar Itihas", Akkhar Publication, Agartala, 1405 Bangabda. Singha, L. Birmangal, & Ray, Pannalal., "Itihaser Aloke Tripura – Manipur, Itihaski Mityengdagi Tripura Manipur", Akkhar Publication, 2007. Singh, W. Ibohal., "The History of Manipur (An Early Period)", Manipur Commercial Co., Imphal, 1986. Sanajaoba, Naorem., "Manipur Past and Present", vol-IV, Mittal publications, 2005. Singha, G.P., "Researches into the History and Civilization of the Kiratas", New Delhi, 2008. Singha, Rajkumar Sri Snahal., (edited) "Chingthangkhomba Maharaja Ganga Chatpa", Sri Noimangthem Kali Singha, Nagamapal, Imphal, 1964. Matrimonial Alliances between the royal houses of Tripura.............. Memchaton Singha Volume-I, Issue-I July 2014 33 Journals/Periodicals Singha, Sri Rajkumar Kamaljit., (edited) "Marup, Sharat ki Echel, 1980", 12 th Edition, Agartala, 1980. Singha, Sri Rajkumar Kamaljit., (edited) "Marup, Sharat ki Echel, 1981", 13 th Edition, Agartala, 1981. "Takhel Lairang", 14 th edition, Manipuri Sahitya Parishad, Tripura, Dec. 2011. Singha, R.K. Jhaljit., "History of Medieval Manipur, part 3", History section, E-Pao, now the World knows (online news paper), September 24, 2010. Website: http://www.epao.com.