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Kawoomba comments on An Intuitive Explanation of Solomonoff Induction - Less Wrong
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As long as our best model of the laws of physics includes the concept of Planck time (Planck length), doesn't that mean that time (space) is discrete and that any interval of time (length of space) can be viewed as an integer number of Planck times (Planck lengths)?
We don't have anything remotely like a well-established theory of quantum gravity yet, so we don't know. Anyway, lack of observable frequency dispersion in photons from GRBs suggests space-time is not discrete at the Planck scale.[http://www.nature.com/nature/journal/v462/n7271/edsumm/e091119-06.html]
Doesn't quite work like this so far, maybe there will be some good discrete model but so far the Plank length is not a straightforward discrete unit, not like cell in game of life. More interesting still is why reals have been so useful (and not just reals, but also complex numbers, vectors, tensors, etc. which you can build out of reals but which are algebraic objects in their own right).
't Hooft has been quite successful in defining QM in terms of discrete cellular automata, taking "successful" to mean that he has reproduced an impressive amount of quantum theory from such a humble foundation.
This is answered quite trivially by simple analogy: second-order logics are more expressive than first-order logics, allowing us to express propositions more succinctly. And so reals and larger numeric abstractions allow some shortcuts that we wouldn't be able to get away with when modelling with less powerful abstractions.