New Foundations for Branching Space-Times

The theory of branching space-times, put forward by Belnap (Synthese 92, 1992), considers indeterminism as local in space and time. In the axiomatic foundations of that theory, so-called choice points mark the points at which the (local) possible future can turn out in different ways. Working under the assumption of choice points is suitable for many applications, but has an unwelcome topological consequence that makes it difficult to employ branching space-times to represent a range of possible physical space-times. Therefore it is interesting to develop a branching space-times theory without choice points. This is what we set out to do in this paper, providing new foundations for branching space-times in terms of choice sets rather than choice points. After motivating and developing the resulting theory in formal detail, we show that it is possible to translate structures of one style into structures of the other style and vice versa. This result shows that the underlying idea of indeterminism as the branching of spatio-temporal histories is robust with respect to different implementations, making a choice between them a matter of expediency rather than of principle.


Motivating a Formal Theory of Indeterminism in Space-Time
The theory of branching space-times (henceforth BST) considers local indeterminism in space and time. Indeterminism is the thesis that a system has more than one alternative possible future evolution. In BST, the emphasis is on the divergence of alternative evolutions occurring in small regions of space and time that can be idealized to be just point events. From the perspective of BST as put forward by Belnap [1], which we will denote BST 92 , indeterminateness gives way to determinateness at point events.
A focus on local aspects of indeterminism is by no means peculiar to the BST 92 analysis of indeterminism. It is quite typical for questions of where and when to come up in indeterministic contexts. We ask, for instance, where and when a person decided on a particular course of action. Or, in a sciencerelated context, we may wonder at which location, and from which instant on, it has been determined that an electron passing a Stern-Gerlach device would be deflected in a given direction. The naturalness of such questions indicates, we believe, that an adequate analysis of indeterminism needs to pay attention to local details of indeterminism: where, when, and how alternative evolutions diverge. These questions are pressing, no matter whether one prefers to analyze indeterminism in terms of branching possible histories, like in BST 92 , or in terms of non-overlapping segment-wise isomorphic possible scenarios of the Lewis tradition.
The question of what the branching of possible histories looks like at the local level is difficult to answer. BST 92 , being a rigorous axiomatic theory, decides the question of how branching occurs via one of its axioms: in BST 92 , the overlap of two possible histories always has a maximal element. The opposite option would be for the difference of two histories to always have a minimal element. Which option is right? Or maybe this cannot be decided, or an answer to the question can be avoided altogether? The present paper shows how to avoid taking sides, in the following sense. First, we present a number of theorems that show that the second option mentioned, branching of histories without maximal elements, can be worked out in a formally precise way. 1 Second, we show that there is a systematic way of translating branching structures of one kind into branching structures of the other kind. In this sense, we can leave it open what branching is really like. For both steps, the notion of a transition structure, which represents local indeterminism, is crucial. 2 A key motivation for working out an ecumenical position is that we do not want to press any global argument for preferring one of the two options mentioned above. We acknowledge that there are valid reasons in favor of both. In the original paper developing BST 92 [1], the decision in favor of maxima in the intersection of histories is commented as follows: Finally, let me explicitly note that on the present theory [. . . ], a causal origin has always 'a last point of indeterminateness' (the choice point) and never 'a first point of determinateness'. I find the matter puzzling since it's neither clear to me how an alternate theory would work nor clear what difference it makes. [1,428] This feeling of puzzlement also stands behind some objections to BST 92 : the objectors ask what the reasons are for assuming one pattern of branching, or they are skeptical whether that pattern is compatible with the physics of space-time. 3 Thus, it seems prudent not to decide the matter of patterns of branching by fiat, at least not in advance of some further considerations of theoretical physics.
A second motivation for this paper stems from topological consequences of preferring one option over the other, and relates to topological requirements standardly imposed on the mathematical structures used in physics for representing space-times, so-called differential manifolds. In this context, it is perhaps worth observing that BST 92 is not meant to be a theory of physics, for two reasons. On the one hand, its axioms are too frugal to be specific enough for a direct application in physics. On the other hand, BST 92 explores the combination of space-time and modality, and that combination does not fall into the standard repertoire of physics. Thus, general relativity represents the evolution of single space-times, whereas branching structures aim to accommodate multiple alternative space-times. In BST 92 one should thus not expect to find the same representational mathematical structures as in physics, since these structures represent different aspects of spatiotemporal reality. But of course, some way of combining these endeavors is desirable. Now, a differential manifold used in physics to represent a space-time has two properties that are hard to satisfy in branching structures. First, by definition, a differential manifold is locally Euclidean, which means that each point of the manifold has a neighborhood that can be mapped continuously onto an open subset of R n (in realistic applications n = 4; see Section 3.2 for the definition). In this way, points in the manifold can be assigned spatiotemporal coordinates via so-called charts. A BST 92 structure, however, is not locally Euclidean with respect to its natural topology (barring trivial one-history cases). The reason is that a neighborhood of a maximal element in the intersection of two histories cannot be appropriately mapped onto R n . We thus face a problem when trying to assign spatio-temporal coordinates to the elements of a BST 92 structure. To address this problem, we will develop a version of BST without maximal elements in the intersection of histories. The natural topology on such structures stands a chance of being locally Euclidean. We are also interested in finding an operation that would transform BST 92 structures into structures that are more friendly to local Euclidicity.
The second property that differential manifolds in space-time physics satisfy, but which is typically violated by BST 92 structures, is a topological separation property known as the Hausdorff property (see again Section 3.2). We will not enforce this feature: the branching structures we develop still violate the Hausdorff property. We will argue, however, that this violation is innocuous, as each separate possible history is Hausdorff in its natural topology. In the same spirit, a manifold representing a single space time of general relativity is Hausdorff. Since we aim at enabling both the assignment of coordinates to events and the representation of alternative spatiotemporal histories, the combination of local Euclidicity and the violation of Hausdorffness on the whole structure seems to be the best result that one can aim to achieve. 4 The paper is organized as follows. In Section 2, we describe the extant theory of BST 92 in formal detail. We comment on topological issues in Section 3. In Section 4, we introduce the "new foundations" BST theory, BST NF . In Section 5 we show how the two frameworks are linked, and we prove general translatability results both ways. We conclude in Section 6.

What is Out There: The Formal Framework of BST 92
Branching space-times theories have been developed in a number of writings starting with Belnap [1]. The dominant theory that has emerged, BST 92 , demands that histories branch at choice points, in the following way: any two histories overlap, and their overlap contains at least one maximal element. That decision has the problematic topological consequence mentioned above, which will be avoided by the novel BST theory to be developed in this paper. There is a substantial common core of the two theories, which we present in Section 2.1. In Section 2.2 we provide some general facts about common BST structures. In Section 2.3 we go on to describe the branching of histories in general terms, and in Section 2.4 we give a formal definition of BST 92 , including its prior choice postulate that demands the existence of choice points.

The Core Theory of Branching Space-Times
In this section we describe the formal core of common BST structures (Definition 2), which is shared by both the established theory of BST 92 and by the "new foundations" theory, BST NF , that we are motivating and discussing in this paper. 5 Both theories are spelled out in terms of partial orderings, so we provide some pertinent general notions first.
. We use the companion notation in the standard way, i.e., e e iff (e < e ∨ e = e ). We extend the ordering notation to sets, with the universal reading, that is: for A set l ⊆ W is a chain iff any two of its members are comparable, i.e., for any x, y ∈ l, either x y or y < x. A set D ⊆ W is (upward) directed iff it contains a common upper bound for any two of its members, i.e., D is directed iff for any x, y ∈ D there is z ∈ D s.t. x z and y z. Definition 2. (Common BST structure) A common BST structure is a pair W, < that fulfills the following conditions: 1. W is a non-empty set of possible point events.
2. < is a strict partial ordering (Definition 1) denoting precedence on W . A 5. The ordering contains infima for all lower bounded chains: If l ⊆ W is a chain that has a lower bound (for some e ∈ W , e l), then l has a unique greatest lower bound inf l, which satisfies ∀x [x l → x inf l.
6. The ordering contains history-relative suprema for all upper bounded chains: If l ⊆ W is a chain with an upper bound (for some e ∈ W , l e), and h ∈ Hist is a history for which l ⊆ h, then l has a unique smallest upper bound sup h l in h: 7. Weiner's postulate: Let l, l ⊆ h 1 ∩h 2 be upper bounded chains in histories h 1 and h 2 . Then the order of the suprema in these histories is the same: 8. Historical connection: Any two histories intersect non-emptily, i.e., for h 1 , From this list, items 1-6 are motivated by the demand that BST structures be continuous orderings such as employed in space-time theories. Item 7 arises as a technical requirement ruling out unintended ordering structures. Item 8, on the other hand, is philosophically motivated by taking indeterminism to be a feature of Our World.

Some Facts About Common BST Structures: Histories and Chains
If our world is indeterministic, then not everything that can happen at all, can happen together-some sets of events are compatible, but others are not. For example, it is possible that it rains in Pittsburgh next week, and it is possible that it rains in Kraków next week, and indeed it is possible that next week, it rains both in Pittsburgh and in Kraków. On the other hand, it is possible that the coin I am about to toss comes up heads, and it is possible that it comes up tails, but it is not possible that it comes up heads and comes up tails. These events are incompatible, or inconsistent. In BST theory, the notion of compatibility is expressed via the definition of a history: histories are maximal sets of compatible events. 6 Formally, as stated in Definition 2(2), a history is a maximal directed set, i.e., a set h maximal w.r.t. the property that for any two events x, y ∈ h, there is some event z ∈ h such that x and y lie in the past of z. A useful motivation for this definition of a history is in terms of the consistency of the past: if there is some possible event z from the perspective of which both x and y have already happened, then x and y are consistent.
Here are some useful facts about histories in common BST structures: As history-relative suprema of chains will play a crucial role in this paper, we provide a number of pertinent definitions and facts.

Definition 3. (Chains and related sets)
We define the following classes of chains and related sets: • C e : the set of chains ending in, but not containing, e. That is: l ∈ C e iff l is an upper bounded chain and there is some h ∈ Hist for which l ⊆ h and sup h l = e, but e ∈ l.
• S (l): the set of all history-relative suprema for an upper bounded chain l: • P e : the proper past of e: We establish the following Facts, using the existence of history-relative suprema and the Weiner postulate: Then there is no history h ⊇ {s 1 , s 2 }.
Proof. Assume otherwise, and let h ⊇ {s 1 , s 2 } for some h ∈ Hist. We have l ⊆ h, since s 1 ∈ h and l s 1 . By Fact 2, we have both sup h l = s 1 (as s 1 ∈ h) and sup h l = s 2 (as s 2 ∈ h). So, contrary to our assumption, s 1 = s 2 .
Here is another useful fact about suprema of chains: If you remove the endpoint of a maximal upper bounded chain, the history-relative supremum does not change. The same holds for infima. Proof. (1) If s ∈ l, we have l = l, and there is nothing to prove. Otherwise, let s = df sup h l . Clearly, l s, so s s (by the definition of suprema). Now assume for reductio that s = s , i.e., s < s. By the construction of l , we then have (*) ∀x ∈ l [x = s → x s ]. By density, there is some e ∈ W for which s < e < s. By (*), we have e ∈ l. But then, again by (*), we have that l * = df l ∪ {e} is also a chain with sup h l * = s, and l * l. This contradicts the maximality of l. So, we have s = s .
The proof for (2) is exactly parallel to that for (1).
The next Fact shows that the proper past of an event e consists of all the chains ending in, but not containing, e. Proof. "⇐" Let x ∈ ∪ l∈C e l, i.e., x ∈ l for some l ∈ C e , and let h ∈ H e . As sup h l = e and e ∈ l, we have l < e, and thus, x < e, i.e., x ∈ P e . "⇒" Let x ∈ P e , i.e., x < e. Then {x, e} is a chain, which by the Zorn-Kuratowski lemma can be extended to a maximal chain l ending in e. Let h ∈ H e ; we have sup h l = e. By Fact 4, for l = df l \ {e} we also have sup h l = e, whereby we have some l ∈ C e for which x ∈ l .

Indeterminism as the Branching of Histories
If a common BST structure contains just one history, then it is trivial from the point of view of indeterminism: all events are compatible, and the picture of a world with just one history is the picture of a deterministic world. Since there are multiple histories in any non-trivial BST structure, there are different ways in which these histories can interrelate. A strong intuitive principle is historical connection (Definition 2(8)): The idea is that any two histories should share some common past. In this way, any common BST structure fulfills Lewis's condition of qualifying as "a world", since it is connected by a "suitable external relation", namely, the relation of precedence, < (see [8], 208).
We will later see that historical connection is implied by stronger principles about the interrelation of histories. These so-called prior choice principles (Definitions 6 and 18) make specific demands on the way in which histories branch off one from another. The key decision is what the branching of histories looks like locally: what are the objects at which histories branch? BST 92 decides for points: histories branch, or remain undivided, at points. With a view to the formal definition of this type of branching in Section 2.4 below, we first provide some essential definitions. We start with the notion of undividedness. Let two histories h 1 , h 2 share some event e ∈ h 1 ∩ h 2 . Then they also may or may not share a later event. In the former case, we call the histories undivided at e: Definition 4. (Undividedness) Let h 1 , h 2 ∈ Hist, and let e ∈ h 1 ∩ h 2 . We say that h 1 and h 2 are undivided at e (h 1 ≡ e h 2 ) iff there is some e ∈ h 1 ∩h 2 for which e < e .
For any event e, the relation ≡ e among the set H e of histories containing e is obviously symmetrical and reflexive, by the form of the definition. We will discuss the issue of transitivity in more detail later on. Our way of enforcing historical connection via prior choice principles will ensure transitivity. Given transitivity, ≡ e is an equivalence relation on H e . We use the notation Π e to indicate the partition of histories from H e into equivalence classes according to ≡ e , i.e., It may be that in fact all histories from H e are undivided at e, i.e., e is not maximal in the intersection of any two histories from H e . In that case, we have Π e = {H e }.
In case two histories h 1 , h 2 share an event e but no event later than e, that event e is a maximum in the intersection of the histories h 1 ∩ h 2 . In that case, we say that the histories split at e: The existence of choice points has important implications for the topological properties of the resulting structures, as we noted above and as we will discuss further in Section 3. At this point, we thus reach an important question: do the postulates of a common BST structure decide whether there are choice points? It turns out that the answer is no: we can show that both the existence and the non-existence of choice points are live options for the branching of histories in common BST structures. Consider, thus, the two common BST structures depicted in Figure 1a, b. 7 These structures illustrate the two possibilities for histories to branch in common BST structures, thus picturing the nuclei of, on the one hand, the well-developed theory of BST 92 (a), and the "new foundations" theory BST NF (b), which will be developed in formal detail in Section 4. 8 We provide a formal definition of these structures, so as not rely solely on pictures. Both structures are defined as quotients of L 2 = df R × {1, 2}, the double real line, under the equivalence relations ≡ a and ≡ b , which are defined, respectively, as These relations differ only in their handling of x = 0. The ordering on the quotient structures M a = df L 2 / ≡ a and M b = df L 2 / ≡ a is defined uniformly via 7 These structures have just the bare minimum of complexity to fulfill the axioms of Definition 2 in a non-trivial way: they contain just two histories each. Furthermore, they do not include any spatial extension-so in fact they are so-called branching time structures as well. 8 The two topological possibilities for branching have been discussed, e.g., by McCall [12], McCabe [11], and Strobach [22,208].
It is easy to check that these structures are non-empty partial orderings without maxima or minima that satisfy the density and continuity (infima and suprema) conditions of a common BST structure. The two histories The intersections of these two histories are, respectively, the upper bounded chains The difference is this: while the chain l a in M a has a maximal element, [ 0, 1 ], the chain l b in M b has no maximal element. That latter chain instead has two different history-relative suprema:

Branching via the Prior Choice Principle of BST 92
Having exhibited the two options for fulfilling the common BST axioms in a simple case, we could now enter into a philosophical discussion of what is the right way to go. We refrain from attempting any a priori arguments here. One can give good reasons for both options. Thus, in favor of the existence of choice points, one can argue that a causal account of indeterminstic choice requires a special last element of indecision, and thus, a maximal element of any two branching histories. In favor of the absence of choice points, one can cite issues of uniformity (it is possible to have branching without maxima in a uniform way), or topological aspects of the resulting structures. Both these issues will be discussed in Section 3. In our view, they provide a good motivation for investigating common BST structures without choice points, and this is what we will do in Section 4. As a matter of fact, however, BST theory was developed with the requirement of the existence of choice points, and the resulting axiomatic theory, BST 92 , has proved to be fruitful for quite a number of applications, e.g., to causation [4], to probability theory [13,25], and to physics [16,17]. So in what follows, we first characterize BST 92 , which is the BST theory with choice points, in full formal detail. This requires the addition of just a single extra axiom to the basis of common BST structures. 9 As originally described by Belnap [1], the theory posits the axioms of a common BST structure together with the so-called prior choice principle, which we will denote PCP 92 to indicate its historical origin. Basically, PCP 92 requires that whenever an event e belongs to one history h 1 but not to another history h 2 , these two histories split at a choice point c in the past of e: It turns out, however, that in order to enforce the transitivity of the relation of undividedness, PCP 92 needs to be formulated not for points, but for lower bounded chains contained in the difference of two histories, as follows. 10 Definition 6. (BST 92 prior choice principle, PCP 92 ) A common BST structure W, < fulfills the BST 92 prior choice principle iff it fulfills the following condition: Let h 1 , h 2 ∈ Hist be two histories, and let l ⊆ (h 1 \ h 2 ) be a lowerbounded chain that is contained fully in history h 1 but does not intersect history h 2 . Then there is a choice point c ∈ h 1 ∩ h 2 s.t. c < l and h 1 ⊥ c h 2 , i.e., c lies properly below l and is a choice point for h 1 and h 2 , which is maximal in the intersection of h 1 and h 2 .
That definition obviously implies the point version described above, as any singleton {e} is a lower-bounded chain. It also ensures historical connection independently of the explicit requirement of Definition 2(8): any two different histories have a non-empty difference (see Fact 1(3)), so that they have to share a choice point.
We can now enter PCP 92 in its official form as an additional item to our list of axioms for BST 92 : Definition 7. (BST 92 structure) A BST 92 structure is a common BST structure W, < (Definition 2) that also fulfills the BST 92 prior choice principle (Definition 6).
In the next section, we continue our description of BST 92 with a focus on its topological aspects. In particular, we provide some facts related to local Euclidicity and Hausdorffness. As indicated, these are the features that are essential for relating histories in BST to space-time structures studied in physics.

Topological Aspects of BST 92
In this section we describe the natural topology for common BST structures, the so-called diamond topology, in Section 3.1. We comment on some of the topological features of BST 92 in Section 3.2. We will return to topological issues for the case of BST NF further down, in Section 4.7.

General Idea of the Diamond Topology
BST admits a natural topology, introduced by Paul Bartha, 11 which we call the diamond topology. The topology is defined either for W , the base set of a BST structure, or for a given history h. In the definitions below, MC(e) (MC h (e)) stands for the set of maximal chains in W (in h) that contain e.

Definition 8. (Diamond topology T on W )
Z is an open subset of W , Z ∈ T , iff Z = W or for every e ∈ Z and for every t ∈ MC(e) there are e 1 , e 2 ∈ t such that e 1 < e < e 2 and the diamond D e 1 ,e 2 ⊆ Z, where Z is an open subset of h, Z ∈ T h , iff Z = h or for every e ∈ Z and for every t ∈ MC h (e) there are e 1 , e 2 ∈ t such that e 1 < e < e 2 and the diamond D e 1 ,e 2 ⊆ Z.
It is straightforward to check that indeed T and T h are topologies, i.e., both the empty set and the base set (W or h, respectively) are open, the intersection of two open sets is open, and the union of countably many open sets is open. The claim of naturalness is based on the observation that these topologies, if appropriately restricted, coincide with the standard open-ball topology on R n , and that the notion of convergence they induce coincides with the order-theoretic notions of infima and suprema. 12 The history-relative topologies are the so-called subspace topologies induced by the diamond topology on W , by taking a history as a subspace of W . This In BST 92 , the global topology and the history-relative topologies have different features. This fact reflects a problem with local Euclidicity, which we discuss next.

Properties of the Diamond Topology for BST 92
We review here some facts about diamond topologies in BST 92 , which are proved in Placek et al. [18]. The first observation is that, unless W, < is a one-history structure, a history h is not open in the global topology T , whereas it is open by definition in its own history-relative topology T h . Generally, if A ∈ T h and A contains a choice point, then A ∈ T , so there is a systematic discrepancy between the global and the history-relative notions of openness. This discrepancy is reflected in a difference with respect to the Hausdorff property, which is defined as follows: Putting aside pathological structures that prohibit the construction of light-cones, 13 it can be proved that the history-relative topologies T h on a BST 92 structure have the Hausdorff property. This fact stands in sharp contrast with the properties of the global diamond topology T : if a BST 92 structure has more than one history, its global topology is non-Hausdorff (again barring pathological structures). Moreover, non-Hausdorffness is related to the existence of upper-bounded chains that have more than one history-relative supremum. As one might expect, a pair of distinct historyrelative suprema of a chain provides a witness for non-Hausdorffness: if any two open sets in T each contain a distinct supremum, they must overlap because they share some final segment of the chain in question.
In physics it is standardly required that individual space-times be Hausdorff (see, e.g., [24], 12). As individual space-times are represented by single histories in a BST 92 structure, we take the above result as showing that BST 92 structures are not in tension with the Hausdorffness requirement of space-time physics. The non-Hausdorffness of the global topology of a BST 92 structure simply reflects the fact that such a structure brings together more than one space-time, explicitly representing a number of alternative spatiotemporal developments.
There is, however, another topological feature of BST 92 that is highly problematic, viz., an issue with local Euclidicity, 14 which is defined as follows: Local Euclidicity is standardly presupposed (often without explicitly mentioning the condition by name) when the notion of a space-time manifold is introduced. On such a manifold, local coordinates are defined via so-called charts (see, e.g., [24], 12f.): at each point of the manifold, it is possible to find a neighborhood that is homeomorphic to some open set of R n , and the respective mapping induces the coordinates. If a topological space is not locally Euclidean, it is not possible to assign coordinates in this way.
Given the frugality of the axioms, BST 92 structures come in many varieties. Hence it is not realistic to hope that their global topology will always be locally Euclidean. One can reasonably require, however, that local Euclidicity should transfer from individual histories to the global structure: if each history-relative topology T h is locally Euclidean, then the global topology T should also be locally Euclidean. If we have some collection of physically reasonable space-times, each with an assignment of coordinates, then a BST analysis of indeterminism should not destroy the coordinate assignment. Unfortunately, local Euclidicity does not transfer from the historyrelative topologies to the global topology of BST 92 . A case in point is the simple two-history model of Figure 1a.  1 ] must extend somewhat to the trunk and to both the arms, i.e., it must con- A fork of that sort, however, cannot be homeomorphically mapped onto an open interval of the real line. Thus, the global topology of M a is not locally Euclidean, despite the fact that each history-relative topology is. Note that no such problem arises for the structure M b of Figure 1b, in which the intersection of the two histories does not have a maximum.

New Foundations for BST, Via Transition Structures in BST 92
Recall that the underlying goal of branching space-times theories is to provide a formal framework for analyzing local indeterminism. We have just seen that one way to achieve that goal, via BST 92 , leads to the failure of local Euclidicity, meaning that there is no way to continuously assign spatiotemporal coordinates to the elements of non-trivial BST 92 structures. Our priority in constructing a "new foundation" theory is to secure local Euclidicity. On the other hand, a violation of the Hausdorff property as in BST 92 , i.e., confined to the global level of indeterministic structures and not already arising at the level of single histories, seems unproblematic. The task we set ourselves is therefore to develop BST theory in such a way that there are no choice points. The resulting formal theory will then provide generalized manifolds, not required to be Hausdorff. On such manifolds one can do calculus and, more generally, develop some space-time physics.
The perhaps surprising fact is that the sought-for framework is readily available via the transition structure of a BST 92 structure. More precisely, starting with a BST 92 structure, we will define the set of its transitions and define an ordering relation on it. The resulting partial order turns out to satisfy all postulates of a common BST structure. However, instead of PCP 92 , it satisfies a different prior choice principle, which, crucially, excludes the existence of maximal elements in the intersection of histories. Quite generally, histories do not split at points, but rather at more complex objects that we call choice sets. Given some mild assumptions, the resulting global structures are provably locally Euclidean.
In this section we introduce the BST 92 notion of a transition (Section 4.1) and show that the full transition structure of a BST 92 structure satisfies all the postulates of a common BST structure (Section 4.2). The resulting notion of a choice set and the emerging pattern of branching are discussed in Section 4.3. In Section 4.4 we formulate a new prior choice principle, PCP NF , and axiomatize the "new foundations" theory BST NF . In Section 4.5 we prove that the postulates of BST NF are satisfied by the transition structure of a BST 92 structure. Some further facts about choice sets are established in Section 4.6. The crucial topological results for BST NF are announced and proved in Section 4.7.

Transitions
The notion of a transition is a powerful tool for discussing indeterminism. Belnap [2] picks up the notion from von Wright [23], adding formal rigor. Generally, a transition is a pair I, O , written I O, in which I is appropriately prior to O, and O is, in some appropriate sense, an outcome of I. Various notions of transitions are discussed in Belnap [4]. For our purposes we focus on the simplest notion of a transition, a basic transition, which in BST 92 is from a possible point event e to one of the immediate possibilities open at e, i.e., from e to a member of the partition Π e of the set H e of histories containing e, Basic transitions are divided up into those that witness local indeterminism, and those at which, so to speak, nothing happens. We denote the set of basic indeterministic transitions of a BST 92 structure W, < by TR(W ), and the set of all basic transitions by TR full (W ).
The set of basic transitions, whether deterministic or indeterministic, admits of a natural partial ordering.
The companion non-strict partial ordering is defined via

Characterizing the Transition Structure of a BST 92 Structure
We are often interested only in indeterministic transitions, as deterministic transitions make no difference to the branching of histories. 15 In the present context, however, it is important to consider all transitions, including the ones that are trivial from the point of view of indeterminism. In a BST 92 model, we therefore define the full transition structure as follows: Definition 14. (The full transition structure of a BST 92 structure.) Let W, < be a BST 92 structure. Then we define the full transition structure (including trivial transitions), Υ( W, < ), using the transition ordering ≺ from Definition 13, as follows: From here on we will denote elements resulting from a transformation with primes.
Having defined the transition structure, we now characterize its properties. It turns out that the full transition structure Υ( W, < ) looks very much like the original BST 92 structure W, < , except for what happens at the choice points. In fact, we will be able to show that apart from the prior choice postulate, all defining properties of BST 92 , i.e., the whole list of properties of a common BST structure from Definition 2, continue to hold; see Theorem 1 below. But by failing the PCP 92 , a transition structure is not a BST 92 structure. With respect to the choice points, the difference is the following. In BST 92 , the branching of histories is from a choice point, shared among the histories that branch, to the immediate possibilities for the future at that choice point. There are no first points in these different possible futures, and this fact leads to the failure of local Euclidicity in the global BST 92 topology (see Section 3.2). In Υ( W, < ), on the other hand, each choice point is replaced by all the transitions that have that choice point as 15 For a study along those lines, see Müller [14].
an initial. Therefore, where in BST 92 there was a last point that was shared between the different possibilities, in the transition structure there are now multiple first points characterizing these different possibilities, and there is no last shared point any more. 16 In the structures of Figure 1, the move from (a) to (b) exactly corresponds to the move from the BST 92 structure M a to its transition structure M b . 17 For the topological consequences, see Section 3.2 above and Section 4.7 below.
We now show that the common BST properties of Definition 2 still hold for the Υ transform of a BST 92 structure. As a first easy fact, we note that Υ( W, < ) is a non-empty partial ordering:   (3)), h contains no maxima either (Fact 1(5)), so there is e 1 ∈ h for which e < e 1 . Accordingly we have τ = df e 1 Π e 1 h ∈ W . It is easy to check that τ ≺ τ , which establishes that τ is not maximal in W .
For no minima, similarly, let τ = e H ∈ W . As W contains no minima, there is e 1 ∈ W for which e 1 < e. Let h ∈ H e . By downward closure, e 1 ∈ h, i.e., h ∈ H e 1 . So there is τ = df e 1 Π e 1 h ∈ W , and τ ≺ τ . Thus, τ is not minimal in W .
The following facts about alternatives to the definition of the transition ordering (Definition 13) will be helpful later on. Proof. (1) We prove the first "iff", from which the second follows immediately. Let e 1 < e 2 . We have to show that H 2 ⊆ H 1 iff H e 2 ⊆ H 1 . As H 2 ⊆ H e 2 , the "⇐" direction is trivial. For "⇒", assume H 2 ⊆ H 1 , and let h ∈ H 2 and h ∈ H e 2 . We have e 2 ∈ h ∩ h , which establishes h ≡ e 1 h . Since h ∈ H 1 ∈ Π e 1 , we have H 1 = Π e 1 h , and by h ≡ e 1 h , h ∈ H 1 as well. So indeed, H e 2 ⊆ H 1 .
In order to prove that Υ( W, < ) is a common BST structure, we need to establish the form that histories, i.e., maximal directed sets, have in that ordering. Their form is quite intuitive, even though it turns out that the proof of that fact is somewhat lengthy. We first establish a useful general fact about directed sets of transitions: Proof. Assume otherwise, i.e., assume that there is some τ * = e * H * ∈ T for which τ 1 τ * and τ 2 τ * . By Fact 7 (1) H(e 3 ) ∈ g for which τ i τ 3 (i = 1, 2), which implies e 3 ∈ E, e 1 e 3 , and e 2 e 3 . This proves that E is directed, and therefore there is some history h ∈ Hist(W ) for which E ⊆ h. We now show that for all e ∈ E, we have h ∈ H(e). Thus, take some e ∈ E, which is the initial of some τ = e H(e) ∈ g. By Fact 6(3) and Fact 1 (5) Given these facts, we can now prove that switching from a BST 92 structure to its full transition structure preserves the common BST structure axioms.

Any lower bounded chain in W , ≺ has an infimum in ≺.
Let l = {e i H i | i ∈ K} (K some index set) be a chain that is lower bounded by e * H * . Then the set l = df {e i | i ∈ K} of initials of l is a chain lower bounded by e * , and there is a history h ⊆ W for which l ⊆ h. By the BST 92 postulate of infima, l has an infimum v in <. The infimum v gives rise to the transition 7. W , ≺ satisfies the Weiner postulate. We will employ the claim established at the end of the previous item (6), and the fact that W, < satisfies the Weiner postulate. Let h 1 , h 2 ∈ Hist(W ), and let h i = {e Π e h i | e ∈ h i } for h i ∈ Hist(W ) (i = 1, 2). We consider two chains l , k ⊆ h 1 ∩h 2 , their respective chains of initials l, k ⊆ h 1 ∩ h 2 , and their history-relative suprema , where s i = sup h i l and c i = sup h i k, for i = 1, 2. Suppose that s 1 c 1 , i.e., s 1 c 1 and Π c 1 h 1 ⊆ Π s 1 h 1 (see Fact 7(3)). By the Weiner Postulate of BST 92 applied to the chains l and k, from s 1 c 1 we may infer s 2 c 2 . Note that s 2 ∈ h 2 . Hence Π c 2 h 2 ⊆ Π s 2 h 2 . In terms of the transition ordering, this means that s 2 c 2 .

Historical connection.
Let h 1 , h 2 ∈ Hist(W ) be histories, which by Lemma 1 correspond to h 1 , h 2 ∈ Hist(W ). By historical connection for W , there is some e ∈ h 1 ∩ h 2 , and by no minima, there is some e * ∈ W for which e * < e. It follows that e * ∈ h 1 ∩ h 2 . Let τ = df e * Π e * h 1 . By Lemma 1, we have τ ∈ h 1 . Now e > e * is a witness for h 1 ≡ e * h 2 , so that Π e * h 1 = Π e * h 2 , i.e., τ ∈ h 2 as well.

Characterizing the Branching of Histories in Transition Structures
We can now discuss formally what has become of the BST 92 choice points, and in which sense a prior choice principle still holds in a full transition structure.
It turns out that when characterizing transition structures, the role of a choice point e in BST 92 is played here by the notion of a choice setë, which describes the local point-wise alternatives for e. In introducing choice sets, we build on some notions introduced above in Section 2.2. We work with a common BST structure W , ≺ derived from a BST 92 structure W, < , i.e., W , ≺ = df Υ( W, < ). Generic elements of W are written e, etc., and histories h, etc. The notions of a chain and of a directed set in this section are relative to the transition ordering ≺. We also call the choice setë the set of local point-wise alternatives for e. Note that e then counts as an alternative to itself. The related notions of alternative histories and history-wise alternatives are defined via the pointwise alternatives:

Definition 16. (Alternative histories and local history-wise alternatives)
We define the set of alternative histories atë, Hë, and the set of local historywise alternatives for e, Πë, to be In order to spell out the variant of the prior choice principle that is appropriate for transition structures, we define two new relations between histories, splitting at a choice set and being undivided at a choice set, written h 1 ⊥ë h 2 and h 1 ≡ë h 2 , resp., in analogy to the respective BST 92 notions. In what follows, we will often use two consequences of Fact 3: 18 Note that Ce = ∅, as W contains no minima.-Our notation with the double dot over e is meant to be suggestive of a number of different history-relative suprema on top of a chain. Think of Figure 1b   Proof. (1) By Fact 3, no history contains more than one history-relative supremum of any upper bounded chain. ( If h 1 ⊥ë h 2 , we say that the choice setë is a choice set for histories h 1 and h 2 . We can establish the following fact about the interrelation of the relations ⊥ë and ≡ë:  To see that this is the same partition as Πë characterized in Fact 9(2), note that H s 1 = H s 2 iff s 1 = s 2 .

Introducing new Foundations: BST NF
With the required notions at hand, we can now propose a new prior choice principle, PCP NF . That principle is crucial for our new foundations: will later show that PCP NF holds in BST 92 transition structures, which themselves are not BST 92 structures (see Lemma 2 below): Definition 18. (PCP NF ) Let h 1 , h 2 ∈ Hist(W ), and let l be a lower bounded chain for which l ⊆ h 1 but l ∩ h 2 = ∅. Then there is some e ∈ W for which e l and for which the setë of local alternatives to e satisfies h 1 ⊥ë h 2 .
Note the weak relation e l in the formulation of PCP NF , in contradistinction to the strict relation in the formulation of the BST 92 PCP in Definition 6. For example, if l has just one element c such that c ∈ h 1 \ h 2 and c is an element of a non-trivial choice setc = {c}, then the choice set for c is justc itself, and h 1 ⊥c h 2 . In such a case we only have the weak ordering relation, c l.
Having proposed a new prior choice principle, we can now give a full definition of "new foundations" BST, BST NF : is a structure of BST NF iff it is a common BST structure (Definition 2) for which the PCP NF (Definition 18) holds.
BST NF structures, being common BST structures, satisfy historical connection just as BST 92 structures do. The new PCP NF , however, implies that the branching of histories in BST NF is different from the branching in terms of choice points in BST 92 : there can be no maximal elements in the intersection of histories in a BST NF structure.
Then h 1 ∩ h 2 , which is non-empty, contains no maximal elements. Accordingly, for any e ∈ W , if e ∈ h 1 ∩ h 2 , then h 1 ≡ e h 2 , where ≡ e is the BST 92 notion of undividedness.
Proof. By historical connection, h 1 ∩ h 2 = ∅. Assume for reductio that there is an e ∈ h 1 ∩ h 2 that is maximal in h 1 ∩ h 2 , i.e., for all e ∈ W , if e < e , then e ∈ h 1 ∩h 2 . By the Zorn-Kuratowski lemma, there is a maximal lower bounded chain J ⊆ {x ∈ h 1 | e x} in the h 1 -future of e. As e ∈ J, we have inf J = e, and by no maxima, J = df J \ {e} = ∅. By Fact 4, e is also the infimum of J . As we have J ⊆ h 1 \ h 2 , by PCP NF there isc with some c 1 , c 2 ∈c such that c 1 ∈ h 1 , c 2 ∈ h 2 , h 1 ∩c = {c 1 } = {c 2 } = h 2 ∩c, and c 1 J . Then c 1 and c 2 cannot share a history (by Fact 9(1)). However, since e = inf J , by the definition of inifima, c 1 e, and as histories are downward closed, c 1 ∈ h 2 . But we have c 2 ∈ h 2 as well, which contradicts Fact 9(1).

BST 92 Transition Structures are BST NF Structures
We can now show that the full transition structure of a BST 92 structure is indeed a BST NF structure. The only thing that is still missing is to show that the new prior choice principle is satisfied. To this end we need an auxiliary fact that shows how BST 92 choice points give rise to BST NF choice sets: in the transition structure, a BST 92 choice point c is replaced by all the basic transitions with initial c, such that these transitions together form a choice set in the resulting structure.  1, 2). In order to show that c 1 , c 2 are elements of a choice setc that fulfills h 1 ⊥c h 2 , we need to show that every chain l ∈ C c 1 , for which sup h 1 l = c 1 , has c 2 as another history-relative supremum, and vice versa. Since W (and thereby W ) has no minimal elements, C c 1 = ∅. Pick an arbitrary chain l ∈ C c 1 , and note that it has the form l = {e Π e h 1 | e ∈ l} for some chain l ⊆ h 1 , with c = sup h 1 l. Since h 1 ⊥ c h 2 , l ⊆ h 2 as well, and as l < c, we have that for every e ∈ l, Π e h 1 = Π e h 2 . Hence l ⊆ h 1 ∩ h 2 . It follows that sup h 1 l = c 1 and sup h 2 l = c 2 (note that the c i ∈ h i are upper bounds of l and that their initial, c, is the h 1 -as well as the h 2 -relative supremum of l). Since l is an arbitrary chain in C c 1 , we showed that every chain in C c 1 has at least two history-relative suprema, c 1 and c 2 , i.e., there is a choice setc Given this auxiliary fact, we can establish our lemma: Lemma 2. Let W, < be a BST 92 structure. Then that structure's full transition structure W , < = df Υ( W, < ) fulfills the new prior choice principle according to Definition 18. Proof. Let l be a chain in W , < that is lower bounded by u , and let h 1 , h 2 ∈ Hist(W ) be such that ( †) l ⊆ h 1 but ( ‡) l ∩ h 2 = ∅. By Lemma 1, Π e h 1 | e ∈ l} for a chain l ⊆ h 1 that is lower bounded by u, where u is the initial of u . By ( ‡), for every e Π e h 1 ∈ l , either e ∈ h 2 , or (e ∈ h 2 but Π e h 1 = Π e h 2 ). There are now three cases, depending on the form of l :   (Fact 1(2)), and in case l has no minimum at all, the assumption that e ∈ l ∩ h 2 implies that there is also some other e 1 ∈ l ∩ h with e 1 < e, and hence e 1 Π e 1 h 2 ∈ l ∩ h 2 , contradicting ( ‡). Thus, l ∩ h 2 = ∅. Applying the PCP of BST 92 to the chain l ⊆ h 1 \ h 2 that is lower bounded by u, we get v ∈ W such that v < l and h 1 ⊥ v h 2 . Exactly like in case (ii) we thus invoke Fact 13 to produce the sought-for choice setv Since v < l and given the form of l , we have v Π v h 1 l as well.
Given this result, we have shown that the full transition structure of a BST 92 structure is a BST NF structure: Theorem 2. Let W, < be a BST 92 structure. Then that structure's full transition structure Υ( W, < ) is a BST NF structure.

Facts About Choice Sets
In this section we prove a few facts related to sets of local point-wise alternatives and sets of local history-wise alternatives, which will also justify our terminology. Our main result is Theorem 3, which states that choice sets fully capture the notion of a local alternative in BST NF . Proof. Assume for reductio that h 1 ⊥c h 2 for some c 1 < e, where c 1 ∈ h 1 ∩c. Then there are c 1 , c 2 ∈c for whichc ∩ h 1 = c 1 = c 2 =c ∩ h 2 . By c 1 < e there is some l ∈ C e with c 1 ∈ l. By h 1 ⊥ë h 2 , we have l ⊆ h 1 ∩ h 2 . But this implies c 1 ∈ h 2 , so that {c 1 , c 2 } ⊆ h 2 , contradicting Fact 9(1).
Lemma 3. Let s ∈ë for some e ∈ W . Then we have x < s iff x < e, i.e., P e = P s .
Proof. If e = s, there is nothing to prove. Thus, assume e = s.
"⇐": Let s ∈ë, and let x < e. By the Zorn-Kuratowski lemma, there is some chain l ∈ C e for which x ∈ l. As s ∈ë, we know that there is some h ∈ Hist for which sup h l = s. We cannot have s ∈ l: otherwise, for h witnessing sup h l = e, we would have {e, s} ⊆ h , contradicting Fact 3. Thus, l < s, which implies x < s.
"⇒": Let s ∈ë, and let x < s. Assume for reductio that x < e. We first show that under this assumption, x and e cannot share a history. Assume otherwise, and let h 1 ∈ H e ∩ H x . Let h 2 ∈ H s . Take some l ∈ C e . We have x ∈ h 1 , x ∈ h 2 (by downward closure of histories, as x < s), and l ⊆ h 1 (as e ∈ h 1 ). Now, as s ∈ h 2 and s ∈ë, we have l < s, so by downward closure of histories, l ⊆ h 2 as well. Noting that sup h 2 {x} = x < s = sup h 2 l, the Weiner postulate implies that sup h 1 {x} = x < e = sup h 1 l, contradicting the assumption that x < e.
Under our reductio assumption, x and e do not share a history. Choose some h 1 ∈ H e and some h 2 ∈ H s . Since s ∈ë and e = s, clearly h 1 ⊥ë h 2 . Moreover, by downward closure of histories we have x ∈ h 2 , and as e ∈ h 1 and x and e do not share a history, x ∈ h 1 . By PCP NF applied to x ∈ h 2 \h 1 , there is c ∈ W such that h 1 ⊥c h 2 and c x, and hence c < s and c ∈ h 2 . And there is c ∈c such that c ∈ h 1 . Picking I ∈ C e and J ∈ C c , I, J ⊆ h 1 ∩h 2 and observing c = sup h 2 J < sup h 2 I = s, the Weiner Postulate implies c = sup h 1 J < sup h 1 I = e. But by Fact 14 then h 1 and h 2 cannot split atc, if they split atë, where c < e. So, on our reductio assumption x < e, we have derived a contradiction, which proves that x < e.
Given Lemma 3 it is also not difficult to see that for s ∈ë, a chain ends in e iff it ends in s: Proof. If e = s, there is nothing to prove. Thus, assume e = s.
"⇐": Given s ∈ë and l ∈ C e , by the definition ofë there is some history h for which sup h l = s, which implies l ∈ C s . "⇒": Let s ∈ë, and let l ∈ C s , i.e., l < s and for some h ∈ H s , sup h l = s. By Lemma 3, l < e. Take some h ∈ H e , and pick some J ∈ C e , so J < e, and hence J ⊆ h . Then we also have J < s (given s ∈ë), which gives us J ⊆ h. We have sup h l = s = sup h J. Thus, by the Weiner postulate, we also have sup h l = sup h J = e, and therefore, l ∈ C e . It follows that the setë is independent of the witness chosen: Proof. Let s ∈ë. We have to show that e ∈s, i.e., e ∈ S (I) for all I ∈ C s . Thus consider an arbitrary I ∈ C s . By Fact 15, I ∈ C e . Now take some h ∈ H e ; we have sup h I = e, i.e., e ∈ S (I). As I was arbitrary, we have e ∈s.
"⇒": Let s ∈ë. For s = e there is nothing to prove, so we assume s = e. "⊆": Let x ∈ë. We have to show that x ∈s, i.e., that x ∈ S (l) for all l ∈ C s . Thus, take some l ∈ C s . By Fact 15, l ∈ C e , and as x ∈ë, we have x ∈ S (l).
"⊇": Let x ∈s. Take some l ∈ C e . As above, by Fact 15, l ∈ C s , and as x ∈s, we have x ∈ S (l).
Having prepared the ground, we can now finally justify calling the partition Πë the set of local history-wise alternatives: The set of sets of histories Πë partitions the set of histories containing P e . That is, any history containing the whole proper past of e ends up in exactly one of the elements of Πë. (2) We have to show that ∪Πë = H [P e ] . Note that ∪Πë = Hë by Fact 9(2). "⊆": Take h ∈ ∪Πë, i.e., h ∈ H s for some s ∈ë. By the definition of P s , we have P s ⊆ h, and by Lemma 3, P e ⊆ h. Thus, h ∈ H [P e ] . "⊇": Take h ∈ H [P e ] , so that P e ⊆ h. By Fact 5, for all l ∈ C e we have l ⊆ h. Take some l 0 ∈ C e , and let s = df sup h l 0 . We can show that s is the h-relative supremum of any chain from C e . Fix some h ∈ H e . Take any l ∈ C e . We have sup h l = e = sup h l 0 , and thus by Weiner's postulate we also have sup h l = sup h l 0 = s. Thus we have s ∈ S (l) for any l ∈ C e , which implies s ∈ë. As h ∈ H s , we have h ∈ ∪Πë.
The main message of the constructions studied in this section is that some e ∈ W generate non-trivial choice setsë, in the sense thatë = {e}. Such a setë indeed consists of local point-wise alternatives to e. We can think of a choice set as a set of "indeterministic transitions", and each choice set induces a set of history-wise alternatives for e, namely Πë. Finally, PCP NF requires that any two histories split at a choice set. So, in BST NF the basic concepts of branching histories still apply, but in a slightly different way from BST 92 . As we will show now, this has the beneficial topological consequences announced earlier.

The Diamond Topology in BST NF
In this section we return to the motivation of this paper, that is, the idea that a framework for local indeterminism should preserve local Euclidicity: if each history (space-time) is locally Euclidean of dimension n, then the global topology should be locally Euclidean of dimension n as well. As we saw in Section 3.2, the diamond topology on BST 92 structures does not preserve local Euclidicity when moving from the history-relative topologies to the global topology. In contrast, we can prove that the diamond topology on BST NF structures preserves local Euclidicity. Working towards Theorem 4 about the preservation of local Euclidicity, we first need an auxiliary lemma, which is also of interest of its own. Recall the disturbing feature of BST 92 discussed in Section 3. Proof. Let Z ∈ T h for some h ∈ Hist. Let e ∈ Z, and let t ∈ MC(e). In order to establish the openness of Z w.r.t T , we need to show that there is an e-centered diamond with vertices on t wholly contained in Z. The openness of Z w.r.t. T h gives us such a diamond for any t h ∈ MC h (e), but not necessarily for our given t ∈ MC(e).
We show that the given t has a segment both below and above e that is contained in some t h ∈ MC h (e), proceeding in two steps. First, we claim that t contains some e ∈ h for which e > e. Assume otherwise, i.e., the chain t + = df {e * ∈ t | e * > e} contains no element of h. Note that by construction, inf t + = e. As t + is a chain, it is directed, and thus wholly contained in some history h 2 . Pulling these facts together, t + ⊆ h 2 \ h, and by the maximality of t and the construction of t + , we have that t + is a maximal chain in h 2 \ h. The PCP NF gives us a choice setc s.t. ( †) h ⊥c h 2 , and for the c ∈c ∩ h 2 , we have c t + . We observe next that from the fact that t + is a maximal chain in h 2 \ h, it follows that c = inf t + . Otherwise, for i = inf t + we would have c < i t + . By ( †) we have c ∈ h, so {c } ∪ t + ⊆ h 2 \ h. As this chain extends t + , it contradicts the maximality of t + in h 2 \ h. Thus, c = inf t + , whence c = e. It follows that e ∈ h 2 \ h, which contradicts our initial assumption that e ∈ h. So indeed, t contains some e ∈ h for which e > e.
Second, we construct t h by starting with an initial segment of the given t, as follows: BST NF thus vindicates the idea that if one starts with locally Euclidean histories (space-times) that allow for the assignment of spatio-temporal coordinates, one does not destroy that feature by analyzing indeterminism within the framework of branching space-times. This is in marked contrast with the situation in BST 92 , in which local Euclidicity does not transfer from the individual histories to the global branching structure. 19 So, if one wants to analyze local indeterminism in branching structures that retain local Euclidicity, one has to choose the framework of BST NF .
With respect to the topological condition of Hausdorffness, on the other hand, the two frameworks are on a par: Apart from some trivial cases, 20 both BST 92 structures and BST NF structures are non-Hausdorff even if their individual histories are Hausdorff.
What is the significance of the failure of transferring topological properties of individual histories to the global structure? Arguably, transfer of local Euclidicity is much more important. Local Euclidicity guarantees the ascription of spatio-temporal coordinates (sets of real numbers) to any spatio-temporal event. The idea that such events have coordinates is deeply entrenched in our concept of space-time. Perhaps there are ways to ascribe coordinates to events that do not require local Euclidicity, but that condition is used in standard physical theories of space-time. Thus, abandoning local Euclidicity will mark a revolutionary break with with the established practice of physics. We thus claim that local Euclidicity should hold both in each history (individual space-time) and in a global BST structure that represents the totality of all possible spatio-temporal events.
The Hausdorff property has a different status. It is standardly postulated to hold in structures representing space-times in General Relativity: theses structures, differential manifolds, are Hausdorff by definition. However, there have been attempts to relax the Hausdorff property, motivated by particular candidates for space-time. It can also be argued that the Hausdorff property is not needed in General Relativity and can be abandoned at a small price [9]. Finally, there is a method of gluing (Hausdorff) differential manifolds into a larger structure, a so-call generalized manifold, that is not Haudorff. It is natural to interpret this result as a modal representation of a family of alternative space-times that overlap on some region [10]. Accordingly, histories should be Hausdorff, but global structures with multiple alternative histories will violate Hausdorffness. 19 Compare our discussion following Definition 11 in Section 3.2. 20 Main examples are one-dimensional structures of BST92 such as pictured in Figure 1. Also, one-history structures are, of course, Hausdorff if their single history is.

From New Foundations to Old Foundations and Back Again
Our target in this section is a set of theorems establishing that we can move freely between BST 92 and BST NF while preserving the basic indeterministic structure. In Section 5.1 we will show that given a BST NF structure, we can define an accompanying BST 92 structure via a transformation detailed in Definition 20, in parallel to the derivation of a BST NF structure from a BST 92 structure above. In Section 5.2, we will then show that the concatenation of these two translations, in any order, is an order isomorphism. In this way, BST 92 and BST NF can be seen as two alternative representations of the same underlying indeterministic structure. This means that we can represent indeterministic scenarios without having to decide between the different prior choice principles of BST 92 and BST NF .

From New Foundations BST NF to BST 92
We have seen how the move from a BST 92 structure to its full transition structure brings us from BST 92 to BST NF . In the other direction, there is also a fairly simple translation, viz., combining, or collapsing, all the elements of a choice set to form a single point. The elements of a choice set constitute different history-relative suprema of a chain without an endpoint. The transform, in contrast, contains a chain with a unique endpoint, after which the different outcomes have no first elements. It will be useful to extend the Λ-notation to elements and subsets of W , so that Λ(e) = dfë , and Λ(E) = df {ë | e ∈ E}. (2) If e 1 < ë 2 and e * 1 ∈ë 1 , e 1 = e * 1 , then e * 1 < ë 2 . So givenë 1 < ë 2 , there is a unique e 1 ∈ë 1 for which e 1 < ë 2 . (3) Ifë 1 < ë 2 , then there are no e * i ∈ë i (i = 1, 2) for which e * 2 < e * 1 .
Similarly to what we established about the properties of the transition structure of a BST 92 structure, we can characterize the Λ-transform of a BST NF structure. It turns out that, as announced, the Λ-transform leads us back to BST 92 . As above, we split the proof into a number of steps.
Proof. (1) By construction, W is non-empty (given that W was nonempty).

Lower bounded chains have infima in < .
Let t ⊆ W be a lower bounded chain, and letb ∈ W be a lower bound for t . The elements of t are of the formë = Λ(e) with e ∈ W . We distinguish two cases. (a) If t has a least element (which covers the case that t has only one element), then that least element is the infimum of t w.r.t. < , by definition. (b) If t has no least element, pick someë ∈ t , and let t * = df {x ∈ t | x < ë}. We have inf t * = inf t by the definition of the infimum. And by Fact 18 (2), for allë 1 ∈ t * there are unique e 1 ∈ W for which e 1 ∈ë 1 and e 1 <ë, and there is a unique b * ∈b for which b * <ë. So there is a unique set t ⊆ W given by which is a chain since t * is a chain; further t is lower bounded by b * ∈ W . By the properties of BST NF , t therefore has an infimum a = df inf t, a ∈ W . We claim thatä is the infimum of t w.r.t. < . As a < t, we havë a < t by the definition of < . Now letc t . Again by Fact 18 (2), there is a unique c ∈c for which c <ë. By the fact that a is the infimum of t, we have c a, which impliesc ä. Soä is indeed the greatest lower bound, i.e., the infimum, of t . 6. Upper bounded chains have history-relative suprema in < .
Let t be an upper bounded chain withb an upper bound, and letb ∈ h for h some history in W , < , so that t ⊆ h as well. As h has a unique pre-image h under Λ (by Lemma 6), alsob and t have unique pre-images b ∈ h and t ⊆ h. So by the BST NF axioms, t has an h-relative supremum s ∈ h. By Fact 17(1), we have t s, and for anyä ∈ h for which t ä, we can consider the unique pre-image a ∈ h ∩ä, for which t a. By the fact that s is the h-relative supremum of t, we have s a, which translates intos ä, i.e.,s is the least upper bound in h , and therefore the h -relative supremum, of t . 7. The Weiner postulate.
Consider two histories h 1 , h 2 ∈ Hist(W ), two chains l , k ⊆ h 1 ∩ h 2 , and their history-relative supremas i = sup h i l andc i = sup h i k (i = 1, 2). Assume thats 1 c 1 . We denote the unique pre-images of h 1 , h 2 , l , k ,s 1 , s 2 ,c 1 , andc 2 under Λ by h 1 , h 2 , l, k, s 1 , s 2 , c 1 , and c 2 , respectively. By the uniqueness of pre-images and properties of < , we have l, Proof. Let h 1 , h 2 be histories in W , < , and let t ⊆ h 1 \ h 2 be a lower bounded chain in h 1 that contains no element of h 2 . We have to find a maximal element c ∈ h 1 ∩ h 2 that lies below t , c < t , and for which h 1 ⊥ c h 2 . The histories h 1 , h 2 have as unique pre-images the W, < -histories h 1 , h 2 . As t ⊆ h 1 , the unique pre-image t ⊆ h 1 . Furthermore, t ∩ h 2 = ∅, for an element e ∈ t ∩ h 2 would give rise toë ∈ t ∩ h 2 , violating our assumption about t . So t ⊆ h 1 \ h 2 . From the BST NF prior choice principle, we have a choice sets and s 1 ∈ h 1 ∩s for which s 1 t, while there is some s 2 ∈s ∩ h 2 . Let c = df Λ(s 1 ) =s; we claim that c is the sought-for choice point. (a) By Lemma 4 we haves 1 =s 2 , and as s i ∈ h i , we haves i ∈ h i (i = 1, 2), so that c =s 1 =s 2 ∈ h 1 ∩ h 2 . (b) As c lies in the intersection of h 1 and h 2 , it cannot be thats 1 ∈ t . This excludes s 1 ∈ t, so that in fact s 1 < t. This in turn implies c =s 1 < t . (c) For the maximality of c in h 1 ∩ h 2 , assume that there isä ∈ h 1 ∩ h 2 for which c <ä. Then we have a unique pre-image a 1 ∈ h 1 ∩ h 2 for which both s 1 < a 1 and s 2 < a 1 , so that both s 1 and s 2 belong to history h 1 . This contradicts Fact 9 (1). So c =s is in fact maximal in h 1 ∩ h 2 . (d) By the definition of ⊥ c , we therefore have h 1 ⊥ c h 2 .

5.2.2.
From BST NF to BST 92 to BST NF Before we can tackle the main Theorem 7, we need to establish an additional fact.
Proof. We claim that we can use the mapping ϕ, defined for e ∈ W 1 to be ϕ(e) = dfë Πë Λ(h) for arbitrary h ∈ H e ⊆ Hist 1 .
(3) Surjectivity: Let a ∈ W 3 . We have to find some e ∈ W 1 for which ϕ(e) = a. As a ∈ W 3 , we have a =ë H, whereë ∈ W 2 and H ∈ Πë . Since W 2 , < 2 is the result of Λ-transform applied to W 1 , < 1 , there is

Conclusion
In this paper we developed a branching space-times theory that constitutes an alternative to the well-studied theory of Belnap [1], BST 92 . We describe both BST 92 and our "new foundations" theory, BST NF , as alternative versions of common BST structures (Definition 2). The difference lies in the way in which histories branch locally, as prescribed by the different prior choice principles, PCP 92 (Definition 6) vs. PCP NF (Definition 18). On the one hand, the difference between BST 92 and BST NF is substantial, as shown by their different topological properties: in BST NF , locally Euclidean individual histories give rise to a locally Euclidean branching structure, i.e., to a generalized manifold, whereas in BST 92 , local Euclidicity does not carry over from individual histories to the whole branching structure. On the other hand, we can prove that both frameworks are formally intertranslatable, so that BST 92 and BST NF can be viewed as different possible representations of a single underlying notion of local indeterminism. In this way, the development of BST NF strengthens the position that branching space-times provides an adequate, formally precise analysis of local indeterminism.