> Suppose that the universe is ultimately discrete at its smallest scales, bounded by some finite length, and everything advances in discrete 'ticks' of time.

> Suppose that the universe is ultimately discrete at its smallest scales, bounded by some finite length, and everything advances in discrete 'ticks' of time. There would be no physical realisation of the real numbers. I'm not a mathematician, or a physicist for that matter - but is that true? You've precluded "infinitely small". But you didn't exclude "infinitely big". If the universe contains infinitely big things, then to it anything finite sized is infinitely small, isn't it? To finish off the thought bubble, as far as I know the universe may end up being infinitely big.