r/badmathematics have already looked at it.

"Being very generous, I think their attempt is to invoke this result of Chaitin to basically say "if the universe was a simulation, then there would be a formal system that described how the universe worked. By Chaitin, there's some 'complexity bound' for which statements beyond this bound are undecidable. But, these statements have physical meaning so we could theoretically construct the statement's analog in our universe, and then the simulation would have to be able to decide these undecidable statements."

What they don't explain is:

• why we should think that we're guaranteed to be able to construct such physical analogs of these statements,
• why they think that whatever universe that is simulating ours must have the same axioms as ours (e.g. Godel only applies to proving statements within the formal system under considerations),
• why they can rule out that the hypothetical simulating computer wouldn't be able to just throw some random value out when it encounters an undecidable statement (i.e. how do we know that physics is actually consistent without examining all events everywhere in the universe?),
• ...or a bunch of other necessary assumptions that they're making and not really talking much about.

They also get into some more bad mathematics (maybe bad philosophy?) by appealing to Penrose-Lucas to claim that "human cognition surpasses formal computation," but I don't think this is anywhere near a universally accepted stance."

The paper is mostly in the "not even wrong territory" so it's hard to offer a concrete refutation, but their arguments apply equally to Conway's Game of Life, a Turing-Complete cellular automata for which Gödel undecidable questions may be asked yet which is easily simulated on a computer.

I'm probably not competent to look at the details, but their paper sets off my BS detector by its reference to Godel's incompleteness theorem and its notion of "non-algorithmic understanding". These are both reminiscent of the Lucas-Penrose idea that consciousness requires uncomputability and that humans have some sort of magical ability to determine the truth-values of Godel sentences. I think the conventional view, sometimes known as the Church-Turing thesis, is that the universe is in fact computable.

Can you disprove the universe is a computer simulation anymore than you can disprove that the world you think you see is just an illusion put into your mind by a demon, as famously mused by the philosopher René Descartes in his Meditations?

In theory, the beings who run the simulation could make whatever they want happen, including change the results of scientific experiments, change the outputs of computers used for mathematics, change the brains of everyone who tries to think about certain mathematical proofs in order to trick them into believing something false, and so on. If you imagine that God or a demon might be tricking you, there is no empirical result and no mathematical or logical result that can prove you're not being tricked.

But of course the question is why we might think the universe is a computer simulation in the first place. No version of the simulation argument or simulation hypothesis has ever made sense. These are two of the objections I think are the most powerful:

• There is no logically valid inference to be made about the world outside the simulation from the world inside the simulation. That's just a logical non-sequitur. The world outside the simulation could be made of pudding for all we know, or it could be made of magic, or it could be run by Zeus. Maybe the simulation is run by a pipe-smoking rabbit and the "computer" is actually a carrot. Why should we be able to say the world outside is any particular way because of the way the world inside is? And if we can't say that, how can we infer there's a world outside, or a simulation, at all?
• There are obvious ethical concerns around simulating universes. It's not at all obvious that, if the advancement of science and technology continued indefinitely, we would eventually simulate a universe like our current universe. I think we would probably outlaw doing such a thing. If you run the simulation and can change it as you like, aren't you responsible for everything that happens in the simulation, especially the things that happen as a result of nature (e.g., infectious disease, natural disasters), and wouldn't that make you the worst murderer of all time?

While checking the Wikipedia page for the simulation hypothesis just now, I noticed the physicist Sean Carroll has a really smart reply to the simulation argument as well:

Of course one is welcome to poke holes in any of the steps of this argument. But let’s for the moment imagine that we accept them. And let’s add the observation that the hierarchy of simulations eventually bottoms out, at a set of sims that don’t themselves have the ability to perform effective simulations. Given the above logic, including the idea that civilizations that have the ability to construct simulations usually construct many of them, we inevitably conclude:

• We probably live in the lowest-level simulation, the one without an ability to perform effective simulations. That’s where the vast majority of observers are to be found.

Hopefully the conundrum is clear. The argument started with the premise that it wasn’t that hard to imagine simulating a civilization — but the conclusion is that we shouldn’t be able to do that at all. This is a contradiction, therefore one of the premises must be false.

This isn’t such an unusual outcome in these quasi-anthropic “we are typical observers” kinds of arguments. The measure on all such observers often gets concentrated on some particular subset of the distribution, which might not look like we look at all. In multiverse cosmology this shows up as the “youngness paradox.”

You could say that maybe the top-level universe is such that it can support infinite computation, so there's never a point when the nested hierarchy of simulations bottoms out. But, in this case, you'd be showing the logic of the first objection I brought up — if we can just stipulate anything we want about the top-level universe, then maybe the top-level universe is a sentient chocolate milkshake, and our universe is just a fleeting dream in its mind.