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 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.
Highlights
Motivating a Formal Theory of Indeterminism in Space-TimeThe theory of branching space-times ( BST) considers local indeterminism in space and time
The theory of branching space-times, put forward by Belnap (Synthese 92, 1992), considers indeterminism as local in space and time
The new PCPNF, implies that the branching of histories in BSTNF is different from the branching in terms of choice points in BST92: there can be no maximal elements in the intersection of histories in a BSTNF structure
Summary
The theory of branching space-times ( BST) considers local indeterminism in space and time. 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 BST92 is not meant to be a theory of physics, for two reasons. The second property that differential manifolds in space-time physics satisfy, but which is typically violated by BST92 structures, is a topological separation property known as the Hausdorff property (see again Section 3.2).
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