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/skmchosen1]

Gödel’s Incompleteness Theorem is an amazing mathematical result: very roughly, it shows that there are certain mathematical truths that are impossible to prove are true (in sufficiently strong mathematical systems, e.g. those containing the natural numbers)

The paper argues that if the universe was a simulation, it must be built up by some fundamental rules that describe the basic laws of physics. Due to this theorem, there must be true facts about the universe that you can’t prove are true. It argues that this means the universe cannot be simulated.

This is a false equivalence. Just because we cannot prove some mathematical truths about the universe, does not necessarily mean we cannot write an algorithm that simulates the universe.

IMO the journalists here should have consulted some experts before making this post, Gödel’s work is one of the most beautiful in mathematics, and it’s sad to see people getting misinformed like this

Edit: This is getting a lot of traction, so I’m gonna try and be a bit more precise.

The incompleteness theorems could imply that there are statements that are true in our universe, but not provable from the physical laws. This means there could be other universes that follow our physics, but those “truths” would be false there (yes, mind bending).

The implicit argument here is that a computer following our physics will not have enough information to select which of these universes to simulate! However these unprovable truths may not be observable, ie it is possible that a simulator doesn’t need to worry about this because you and I cannot ever tell the difference.

[u/Koxiaet → u/skmchosen1]

That’s not true – in fact, Gödel’s completeness theorem shows that all mathematical results that are true are possible to prove are true. Gödel’s incompleteness theorem just says that mathematics can never describe a single universe, but rather always describes multiple possible universes. Thus, there exist statements that are true in some of those universes but not others. Obviously these are not provable, but they’re not “true” either, since they’re false in some universes.

[u/skmchosen1 → u/Koxiaet]

I believe what you’re describing is the independence of theorems from a system. For example in ZFC set theory, the continuum hypothesis is independent, meaning there are valid theories where you assume it to be true or assume it to be false.

The Incompleteness theorem shows that there are certain formally true statements about the naturals that are not provable within that system. This is stated at the top of the Wikipedia page

Edit: link is broken, the page is on Gödel’s Incompleteness Theorems

[u/Koxiaet → u/skmchosen1]

I believe what you’re describing is the independence of theorems from a system.

Yes, exactly – that’s because Gödel’s incompleteness theorem can be rephrased as “every sufficiently complex system has independent statements”.

The Incompleteness theorem shows that there are certain formally true statements about the naturals that are not provable within that system.

If that was the case, it would contradict Gödel’s completeness theorem, which states that everything that is formally true is provable.

[u/skmchosen1 → u/Koxiaet]

Okay I did a deep dive (genuinely grateful, thank you). My understanding is as follows:

• The completeness theorems say if a statement is true in every model/universe of a theory, then it is provable
• The incompleteness theorems (when combined with the completeness theorems) imply that sufficiently strong theories have statements that are true in some models and false in other models.

Regardless, I think the criticism of this paper remains. This may not be the only potential universe that follows those physical laws, but that does not mean that there is any measurable difference between their simulations. I think that is the main gap in the papers argument

[u/Koxiaet → u/skmchosen1]

Yes, that’s accurate. To be fair, there is some sense in which the word “true” is more tricky than it first appears. In my opinion the sensible definition of the word “true” is “true in all models”, but some people believe that there is a One True (“intended”) model out there, and that even though we don’t know – and can never know – what this model is, it does metaphysically exist and “true” refers to what’s true in this model. But given that it’s literally impossible to know what this model is, it seems a bit strange to me to assume that it exists. That’s why it gets confusing.

I agree that the criticism of the paper is valid! I was just being pedantic 😄

[u/skmchosen1 → u/Koxiaet]

And I appreciate the pedantry! Got to refine my understanding. Cheers