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I've read that when Turing-Church thesis is applied to the universe and physics, one of the three interpretations that we can use and is defended by some important physicists is that:

"The universe is a hypercomputer and then it is possible to build more powerful machines than Turing machines. For this it would be enough for the universe to be continuous and make use of that continuity (another question is how dense its continuity is), using the results of said supercomputer as input"

Would that mean that every continious (or "continously-enough") model of spacetime and the universe could have fundamentally hypercomputational physics (describing a hypercomputer-like universe)? Or on the contrary only certain models could do it? In that case, can you think of any in particular?

Links

Turing-Church thesis

(https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis)

(https://es.wikipedia.org/wiki/Tesis_de_Church-Turing)

Hypercomputation

(https://en.wikipedia.org/wiki/Hypercomputation)

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  • $\begingroup$ The gotcha is in the phrase, "make use of the continuity." The world outside your window is continuous as far as you can tell, but you can't even start down the road towards using it to hypercompute because so many systems are dominated by randomness below a certain level, and others are linear, and so on. The continuity would have to be beyond "not discrete," it would have to reach "not even possible to approximate discretely." $\endgroup$
    – Display Name
    Feb 20, 2019 at 22:39

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All the laws of physics we know of can be approximated as discrete systems, even though they are written down as if everything was continuous. A continuous reality would not get you past the Turing machine unless it couldn't be well-approximated by a Turing machine to an arbitrary precision. In that sense being continuous wouldn't be enough (otherwise we would consider $\pi$ a "hyper-number,"). If you wanted it to hyper-compute, the universe would have to be so continuous, and have so many macroscopically significant weird things wrapped up at all length scales, that a discrete approximation wouldn't work.

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