Causal sets and frame-valued set theory

In spacetime physics any set C of events—a causal set—is taken to be partially ordered by the relation ≤ of possible causation: for p, q ∈ C, p ≤ q means that q is in p’s future light cone. In her groundbreaking paper The internal description of a causal set: What the universe looks like from the inside, Fotini Markopoulou proposes that the causal structure of spacetime itself be represented by “sets evolving over C” —that is, in essence, by the topos SetC of presheaves on Cop. To enable what she has done to be the more easily expressed within the framework presented here, I will reverse the causal ordering, that is, C will be replaced by Cop, and the latter written as P—which will, moreover, be required to be no more than a preordered set. Specifically, then: P is a set of events preordered by the relation ≤, where p ≤ q is intended to mean that p is in q’s future light cone—that q could be the cause of p, or, equally, that p could be an effect of q. In that case, for each event p, the set p↓ = {q: q ≤ p} may be identified as the causal future of p, or the set of potential effects of p. In requiring that ≤ be no more than a preordering—in dropping, that is, the antisymmetry of ≤—I am, in physical terms, allowing for the possibility that the universe is of Gödelian type, containing closed timelike lines.
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