Now let the symbol IR³(O) stand for the set of (unpossessed) exact positions relative to some object O. Since no object ever has a sharp (mathematically exact) position (relative to another object)...

In a non-relativistic world this is so because the exact localization of a particle implies an infinite momentum dispersion, which in turn implies an infinite mean energy. In a relativistic world the attempt to produce a strictly localized particle results instead in the production of particle-antiparticle pairs.

...we can conceive of a partition of IR³(O) into finite regions that are so small that none of them is the sensitive region of an actually existing detector. Hence we can conceive of a partition of IR³(O) into sufficiently small but finite regions R_i (i=1,2,3,...) of which the following is true: there is no object Q and no region R_i such that the proposition "Q is in R_i" has a truth value. (The possible truth values, recall, are "true" and "false".)

In other words, there is no object Q and no region R_i such that R_i (or the distinction we make between "inside R_i" and "outside R_i") exists for Q.

But a region of space that does not exist for any material object, does not exist at all. The distinctions we make between the regions R_i are distinctions that Nature does not make. They correspond to nothing in the real world. They exist only in our heads.

It follows that the spatial differentiation of the physical world is incomplete. It doesn't go all the way down. If in our minds we partition the world into smaller and smaller regions, there comes a point when there isn't any material object left for which these regions, or the corresponding distinctions, exist.

What about the world's temporal differentiation? Like the properties of physical systems or the values of observables, the times at which properties or values are possessed must be indicated ("measured") in order to exist. As detectors are needed to both indicate and realize positions, so clocks are needed to both indicate and realize times.

Clocks realize times by the positions of their hands. Since exact positions do not exist, neither do exact times. Like the existing spatial relations, the existing temporal relations are fuzzy.

Digital clocks indicate times by transitions from one reading to another, without hands. The uncertainty principle for energy and time, however, implies that such a transition cannot occur at an exact time, except in the unphysical limit of infinite mean energy.

From the fuzziness of temporal relations the incomplete temporal differentiation of the world follows in exactly the same way as its incomplete spatial differentiation follows from the fuzziness of spatial relations.