New Foundations for Branching Space-Times

Abstract

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 spacetimes 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.

Семантика Переходов И Детерминизм Во Временной Логике

“…На текущий момент можно говорить о том, что исследование проблематики детерминизма в рамках логики ветвящегося времени расходится на два самостоятельных направления. Первое из них связано с дальнейшими работами Бэлнапа и предложенным им проектом ветвящегося пространства -времени (branching space-time) (Belnap, 1992;Belnap et al, 2021a;Belnap et al, 2021b;Belnap & Placek, 2012: ). Общей идеей данного концепта можно считать попытку совместить представления об индетерминизме с положениями релятивистской физики.…”

Branching with a Humean Face

“…Another reason for doing so is that it seems to me that some of the finer formal aspects are not essential to the philosophical merits of the theory, and rigorous alternatives with a similar general outlook have been considered. Some of those have found way toBelnap et al (2021), which offers "new foundations" for the theory, but with which the philosophical community seems yet to have engaged. They also feature in the 2022 monograph by the same authors.…”

“…The technical paper(Belnap et al, 2021) gives the BST theory its "new foundations", which are used extensively in the 2022 monograph. In particular, the new approach does away with choice events, switching to "choice sets", thanks to which the theory has better topological features and becomes compatible with the manifold topology of GR.…”

On the Cardinality of Many-Worlds in Discrete Spacetime Structures

“…Consequently with our version of causal sets, under appropriate assumptions on the properties of the universe, one can describe light cones as finitely branching tree structures and define worldlines as infinite paths on such trees or equivalently as members of 2 N Cantor space. 2 Using some facts about effectively closed subsets of Cantor space, we observe that for any event x, if all worldlines extending x are 'eventually deterministic', in the sense that there exists an event above which there is no more split, then the number of alternate universes with respect to x is exactly ℵ 0 . Furthermore, if there are countably many alternate universes with respect to x, then at least one of them must be decidable in the sense that there is an effective uniform method, i.e.…”

“… See Belnap, et al (2021) [2] for the revised version of branching space-time 2. In fact we will use the space n N for n ∈ N, which we shall refer to it with the same name…”