Consequences for Physics

Is the entire past and future history of our universe describable by a finite sequence of bits, just like a movie stored on a compact disc, or a never ending evolution of a virtual reality determined by a finite algorithm, such as in Example 2.2? Contrary to a widely spread misunderstanding, quantum physics, quantum computation (e.g., [3]) and Heisenberg's uncertainty principle do not rule this out [35]. In absence of contrarian evidence we might assume our universe is formally describable indeed, or at least sampled from a formally describable distribution -- if this assumption is false, then our world will forever remain beyond formal understanding.

As obvious from equation (13), the future of some observer evolving within such a universe depends on this prior distribution. More or less general notions of TM-based describability put forward above lead to more or less dominant priors such as on formally describable universes, and on enumerable universes, and and recursive priors on monotonically computable universes. Generally speaking, the theorems above show that any future corresponding to a history without any short description (given the appropriate TM type) is necessarily unlikely. To a certain extent, this justifies ``Occam's razor'' (e.g., [4]) which expresses the ancient preference of simple solutions over complex ones. A more detailed analysis can be found elsewhere [36].