https://arxiv.org/pdf/2404.16050

What the author did was use the Physical Church-Turing thesis, and Kleene's second recursion theorem, to show that: (1) If a universe’s dynamics are computable (PCT), and (2) the universe can implement universal computation (RPCT), then (3) the universe can simulate itself, including the computer doing the simulating.

That's basically all. And thus "there would be two identical instances of us, both equally 'real'." (Two numerically distinct processes are empirically identical if they are indistinguishable. You might remember this sort of thing from late 20th c. philosophy coursework.)

He also uses Rice’s theorem (old) to show that there is no uniform measure over the set of "possible universes."

It's all very interesting, but it's more a review article than a "new mathematical framework." The notion of a mathematical/simulated universe is as old as Pythagoras (~550 BC), and Rice, Church-Turing, and Kleene are all approaching the 100-year mark.

If the universe isn't "real" in the materialist sense, that does not imply that there's a "real" universe outside of the one we perceive, nor does it imply that we're being "simulated" by other intelligences.

The path of minimal assumptions from reality not being "real" is idealism. We're not simulated, we're manifesting.

The classical naive way of obtaining a consistent causal chain, is to put the links one after the other following the order defined by the simulation time.

The funnier question is : can it be done another way ? With the advance of generative AI, and things like diffusion model it's proven that it's possible theoretically (universal distribution approximation). It's not so much simulating a timeline, but more sampling the whole timeline while enforcing its physics-law self-consistency from both directions of the causal graph.

In toy models like game of life, we can even have recursivity of simulation : https://news.ycombinator.com/item?id=33978978 unlike section 7.3 of this paper where the computers of the lower simulations are started in ordered-time

In other toy model you can diffusion-model learn and map the chaotic distribution of all possible three-body problem trajectories.

Although sampling can be simulated, the efficient way of doing it necessitate to explore all the possible universes simultaneously like in QM (which we can do by only exploring a finite number of them while bounding the neighbor universe region according to the question we are trying to answer using the Lipschitz continuity property).

Sampling allows you to bound maximal computational usage and be sure to reach your end-time target, but at the risk of not being perfectly physically consistent. Whereas simulating present the risk of the lower simulations siphoning the computational resources and preventing the simulation time to reach its end-time target, but what you could compute is guaranteed consistent.

Sampled bottled universe are ideal for answering question like how many years must a universe have before life can emerge, while simulated bottled universe are like a box of chocolate, you never know what you are going to get.

The question being can you tell which bottle you are currently in, and which bottle would you rather get.

How is this consistent with the second law of thermodynamics? If there is one universe containing an infinite number of simulations (some of which simulate the base universe) wouldn’t there be a limit to how much computation could be contained? By its very nature a chain of simulations would grow exponentially with time, rapidly accelerating heat death. That may not require the simulations to degrade but it puts a hard limit on how many could be created.

Right; that's the feeble public imagination. What captures my imagination is the idea that the existence of the rules alone is enough to obtain the universe; no simulator is required.

We can make an analogy to a constant like pi. No division has to take place of a circumference by a diameter in order to prop up the existence of pi.

The requirement for a simulator just punts the rock down the road: in what universe is that simulator, and what simulates that? It's an infinite regress. If there is no simulator, that goes away.

If certain equations dictate that you exist and have experiences, then you exist and have experiences in the same way that pi exists.

A non-realtime simulation would allow you certain solutions (such as perfectly recreating a past state of the current universe), but might not allow you to practically see a future state.

So, as long as one believes in continuum, this is just toying around?

However there is no way "you can physically isolate isolate your laptop, from the rest of the Universe" so doesn't that refute this example (at least?)

Many would expect that the parent universe would be more sophisticated, potentially with more dimensions, that we can only glimpse through artifacts of the simulation.

> Consequences of Undecidability in Physics on the Theory of Everything

https://news.ycombinator.com/item?id=45770754

By the way, on a related note, I once stumbled across a paper that argued that if real numbers where physically realizable in some finite space, then that would violate the laws of thermodynamics. It sounded convincing but I also lacked the physical knowledge to evaluate that thesis.

bad bad not good.