I’ve thought a lot about the physical meaning of infinity, but it just dawned on me that a simple thought experiment suggests that infinity could be physically meaningful, in the simple sense of unbounded quantity: just consider the unbounded expansion of time. There’s no reason to assume that the Universe will ever vanish, even if all the mass and energy within the Universe does vanish. This posits a vacuum that exists, and persists through time, implying time is infinite, even if there’s nothing happening in that vacuum (i.e., it’s just an empty space that nonetheless exists).
Let’s dispense with that notion, and consider instead the mass and energy in the Universe itself. Fields can effectively destroy energy – just imagine light escaping from a massive object, it could red shift into literally nothing, if energy is quantized. However, fields cannot (to my knowledge) destroy mass. As a consequence, if all mass and energy were to vanish, it would have to be the case that all mass is converted to energy, and then destroyed via gravity. However, you can’t have gravity without mass, which implies that it is literally impossible for all of the energy in the Universe to be destroyed, since even in this view, you’d always have at least some mass. Therefore, time must be infinite, absent other assumptions.
If time is infinite, then it suggests the more general possibility of unbounded quantities, in particular, arbitrarily precise measurements. You don’t need the set of real numbers to have problems with predictions, you just need arbitrarily precise measurements, which would always imply non-zero error (i.e., the unexplored portion could always contain information). If the consequences of the error are meaningful, your predictions will be meaningfully wrong. Because humans have finite lives, if arbitrary precision is real, then it necessarily implies all predictions carry uncertainty.
Information Overload 