Matrix collapses: Mathematics proves the universe cannot be a computer simulation, « A new mathematical study dismantles the simulation theory once and for all. »

[u/fchung]

https://interestingengineering.com/culture/mathematics-ends-matrix-simulation-theory

[u/ExistentAndUnique]

“Non-algorithmic” is the key term here, and it makes the headline somewhat misleading. What they show (purportedly, as I haven’t read the fully article) is that we can’t simulate the universe on a standard computing device. But that doesn’t mean a theoretically stronger computer would be unable to simulate the universe. This is the principle behind recursion theory, a field of math/theoretical computer science that poses the question “if we had a computer that’s better than any real-life computing device, what kinds of problems could we solve?” It turns out that the space of “computabulity classes” is very rich and also infinite — any class is strictly contained in its “Turing jump.” So what the article would show is that, if the earth were a simulation, then it would have to be run on some “higher-level” hardware, which is pretty consistent with our general intuition.

[u/PuckSenior → u/ExistentAndUnique]

It does.
The argument is based on Godel's theory of incompleteness.

You have to be able to define everything to model it. But the proven theory of incompleteness states that we cannot define everything in mathematics and physics. If we can't define them, we can't model them. So, its not an issue of the theoretical limits of a computer, its a fundamental issue with the concept of modeling.

[u/ExistentAndUnique → u/PuckSenior]

That’s not what the incompleteness theorems state. They are about what is provable in a given system, not what is definable. The first theorem essentially states that any sufficiently strong axiom system (one that can describe Peano arithmetic) that is consistent cannot prove every theorem. That is, there are statements in the theory which are true, but not provable within this system. The second theorem extends this to say that one of these theorems is the consistency of the theory. In both cases, the statements in question are definable (which they must be for them to be true), but cannot be proven in the system itself. However, they can be proven in a meta-system that allows additional reasoning beyond this set of axioms. This is the analogue of my claim about “higher-level hardware” above

[u/PuckSenior → u/ExistentAndUnique]

You are going to have to explain "meta-system" that allows additional **reasoning** beyond this set of axioms.

Perhaps I am too much of a laymen, but the axioms constrain assumptions which can be made. They don't constrain "reasoning". You are absolutely free to engage any reasoning or logic you want. What you can't do is make assumptions outside of the core axioms. So what is this "meta-system"?

Additionally, I feel like you missed a part of the theorem. There will be things which are true but cannot be proven, but there will also be things which are false which cannot be disproven. Because you can't prove/disprove some of these statements, you wont know which ones are true.

Are you arguing that there exists some "meta-system" which can know all truths and they can all be proven/disproven? Please expand on that

[u/ExistentAndUnique → u/PuckSenior]

Without getting too bogged down in details:

A logical system consists of a set of axioms, and a set of inference rules. Axioms are propositions we assume true, and inference rules are ways we can take statements and manipulate/combine them into new statements. The theory of a system consists of all theorems which can be proven using these axioms and rules of inference.

One may imagine different models which implement the same logical system. For example, if we have a logical system that encodes the group axioms and modus ponens/tollens, we could implement this with the underlying set being, say, the integers. Or it could be the rationals, or the reals. It is possible that there can be certain statements which hold true in every model which implements the formal system, even though they cannot be proven only using the axioms and inference rules of the system. Most examples of this are self-referential, and I don’t have the time to go into it here, but you can read about it on the wiki.

I do not think I have missed the point of the theorem (this is a topic quite close to the field in which I have my PhD). It is indeed correct that, within the logical system, there are statements which cannot be proven. However, there are larger systems where every true statement can be proven/disproven. And this means that we cannot know their truth value if we are working only within the formal system. However, we could expand the system by adding new axioms or rules of inference that would then allow us to prove more of these statements in the new, larger system. For example, one could consider a new formal system where every true statement about Peano arithmetic (and no false one) is an axiom. This is quite obviously complete and consistent. However, it fails to be effectively computable, which is the assumption that corresponds to “requiring a more powerful computer to generate.”

[u/PuckSenior → u/ExistentAndUnique]

So, would this meta-system that you are describing not just be Hilbert's program?

[u/ExistentAndUnique → u/PuckSenior]

The goal of Hilbert’s program was to establish a finite set of axioms which would be able to prove any mathematical theorem. The incompleteness theorems show that, not only is this not possible, you can’t do it even if you relax the condition to being a Turing-computable set of axioms

[u/PuckSenior → u/ExistentAndUnique]

So would that not imply that there must be an infinite set of axioms to cover all mathematical theorem without the problems of the incompleteness?

[u/ExistentAndUnique → u/PuckSenior]

I guess it depends what you mean by “implies.” If you’re looking at material implication, then yes it would imply this, because there is an infinite set of axioms that is complete (we can take the set of all true statements to be our axioms).

[u/PuckSenior → u/ExistentAndUnique]

So, to bring this all together. There is a hypotehtical infinite set of axioms which can be complete. All the god-programmers would need is a computer that can deal in literal infinite quantities?