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Could the universe be modeled or thought of in terms of an countably infinite state machine? Philosophically, I am asking from a deterministic perspective. Now, I know that as humans, and beings within our closed system (the universe) it is impossible for us to know the velocity and position of any particle, but my question is more about if determinism is true. And, if it is true, would it fit a countably infinite state model? I am researching things such as Planck's time, and Planck's length, and it is not clear as to if the current state of the universe (position and velocity and perhaps other variables) changes given an increment of time less than Planck's time.

for, if the position and velocity of particles does indeed change given time less than Planck's time, then it seems to me that the states of the universe cannot be mapped to countably infinite states. Meaning, the number of states is uncountably infinite rather than countably infinite. Like the distinction between real numbers and integers.

I am simply a confused layman, who would like some insight.

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    $\begingroup$ A 'countably infinite state machine' is a Turing machine, unless you constrain it further. $\endgroup$
    – user107153
    Commented Aug 2, 2017 at 20:31
  • $\begingroup$ Yes that is true. I could have just said a Turing machine, but my question is the same either way. I simply wanted to emphasize the infinite states aspect of turing machines. $\endgroup$
    – Jacob Levinson
    Commented Aug 2, 2017 at 20:46
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    $\begingroup$ I think you then want to look at stuff about the computability of physics. This has a well-defined but slightly non-obvious definition which is to do with being able to compute successively better approximations with a countable-state (Turing-)machine to a continuous system. I believe that all of physics is thought to be computable in this sense, and I also think that this means that yes, you can so model the universe. Penrose would differ! $\endgroup$
    – user107153
    Commented Aug 2, 2017 at 21:43
  • $\begingroup$ Even if non-Planckian possible QFT states are countable (like eg the finite entropy of a black hole which is the log of Planckian volumes in the horizon), we don't know how many sub-Planckian states there are.String Theory, eg, you can have infinitely countable set of (eg oscillation) states for a string or n-brane. But sTring Theory sort of assumes quantum theory. Our current quantum gravity theories (eg. AdS/CFT) are countabley infinite states. They are all 'quantized'. But we don't know the real theory. Could involve freaky effects that don't map to the integers, not quantized. Not Turing. $\endgroup$
    – Bob Bee
    Commented Aug 3, 2017 at 5:33
  • $\begingroup$ It can be modeled by, but cannot be modeled as, a Turing machine, unless you abandon relativity and quantum mechanics. $\endgroup$
    – Mitchell Porter
    Commented Nov 23, 2019 at 2:10

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The distinction between a model of the universe that contains a true continuum (and thus infinite possible states) and models where that continuum is ultimately discretized sounds very appealing at first glance, but it's ultimately not that useful. As a simple example, what if the world can inhabit a continuum of states, but no experiment can distinguish, even in principle, states that are too close together; are those states still "different"? There is indeed quite a lot that you can do to formalize those notions, but it takes a fair amount of work.

That said, what you're looking for is probably what's known as the Bekenstein bound, which gives fundamental limits on the information that a region of space can contain, via Planck-physics considerations like those in your question, and which is ultimately proportional to the surface area of the region. Scott Aaronson just did an excellent job of explaining this in an accessible way in his past Is "information is cal" contentful, so i won't try to compete, but I do want to know that the bounds he discusses really are (essentially) the best we can do with the physics we know at the moment.

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  • $\begingroup$ Yes, but that assumes there is no subPlanckian degrees of freedom. Only after we get an accepted quantum gravity theory could we maybe know. As I noted in my comment above, sub-Planckian physics is a current research area $\endgroup$
    – Bob Bee
    Commented Aug 3, 2017 at 5:36
  • $\begingroup$ In a closed universe is the number of bits finite? $\endgroup$
    – Keith McClary
    Commented Nov 23, 2019 at 5:07

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