Dear Lawrence,
thank you very much for your further comments!
I totally agree that we cannot have direct information on transfinite sets. Information is discrete and computable, while infinite sets contain mostly uncomputable elements. But I'm not sure that some transfinite sets cannot have a physical meaning. Of them it is possible to have discrete models, as is done for example for real numbers. Moreover we don't know what space and time are made of. I know that the prevailing view nowdays is that they have a discrete and quantum structure. But it is not necessarily so. They may be continuous and composed of uncountable set of points (space) and instants (time). Important authors (as Russell and Grünbaum) have argued that only by admitting the continuum and the actual infinity we can solve Zeno's paradoxes.
http://www.iep.utm.edu/zeno-par/
Cheers, Giovanni