Causal Sets: Geometry and Inference

Is there a time shorter than a second? It's a question a child might ask. We can answer. Half a second. A quarter of a second. Most sophisticated adults, trained by a modern education system, probably think a smallest quantity of time is as absurd an idea as a smallest number that's bigger than zero.
On reflection, though, this seems hubristic. Numbers are not real – they are conceptual tools. Space and time are aspects of reality. They should not be expected to follow the same idealised rules as abstract concepts like numbers. We don't expect that other real objects can be infinitely subdivided: the atom has been split; the protons and neutrons turned out to be made from quarks; but nobody really thinks we can keep breaking up these material objects forever. So why should we think the time and space in which these objects exist can be infinitely subdivided?
Heisenberg's Uncertainty Principle tells us that we cannot know an object's position or its momentum precisely. It is related to the "Planck distance", an unimaginably small quantity, about a septillionth of the diameter of a hydrogen atom. At this scale, quantum effects such as uncertainty dominate and the law of gravity becomes profoundly incompatible with the laws of quantum mechanics. The concepts of length and duration become less meaningful; measurements no longer have much to do with our familiar universe. Around these tiny scales, the child's question doesn't exactly find an answer, but it does reveal that our comfortable understanding of infinite subdivisibility of the universe might be a bit simplistic.
Einstein's geometry assumes infinite subdivisibility, and this is essential to the powerful tool of calculus which describes the law of gravity. However, in a universe where quantum mechanics reigns, infinite subdivisibility cannot hold true. On the other hand, whatever geometry we use to make sense of the counterintuitive quantum world, when we zoom back out to human scales, or massive...