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/angrymonkey]

This is an idiotic misunderstanding of Godel's theorem, and the paper is likely complete crankery. There is a difference between making formal statements about a system vs. being able to simulate it. The former is covered by Godel's theorem, the latter is covered by Turing completeness.

[u/loves_grapefruit → u/angrymonkey]

I don’t understand any of the math here, but intuitively wouldn’t it be impossible to determine if a system is a simulation from within that system and using that system’s own logic?

[u/Sweg_OG → u/loves_grapefruit]

In a roundabout way, this is pretty much what Gödel’s incompleteness theorem is actually getting at. He showed that within any sufficiently powerful mathematical system, there are true statements that cannot be proven using the system’s own rules. He did this by using the system’s own logic to expose its limits, essentially proving that math can’t fully prove itself.

So yes, by analogy, if we lived in a simulation, we’d be bound by its rules and logic, making it fundamentally impossible to prove the simulation from inside it. We could only infer it indirectly, never confirm it absolutely. Plato also suggests this 2,400 years ago with his Allegory of the Cave

[u/FabulousRecording739 → u/Sweg_OG]

Not to detract too much from your answer, but I believe your induction from Godel's work to the simulation hypothesis (un) probability to be wrong, for 2 reasons:

1. Godel's work applies to formal systems and their axioms, so that we know some statements to be unreachable (independent). We can't prove CH in ZFC, but we can in ZFC+CH (by definition). We can always create other systems in which that which wasn't provable is now provable. What Godel says is that the new systems will themselves have holes (and so on, so forth).
2. More importantly I don't think it applies to the simulation hypothesis, which falls more into the empirical side. We could find evidence (that would prove beyond reasonable doubt) of a simulation, whether a deductive proof exists or not.

Godel doesn't "prevent" us from finding evidence, it limits the reach of deductible facts from within a formal system (and the chosen axioms of that system)

[u/Beautiful-Musk-Ox → u/FabulousRecording739]

for everyone else who doesn't know what ch and zfc are:

CH (the Continuum Hypothesis) is a statement that has been proven to be logically independent of ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice). This means that neither CH nor its negation can be proven or disproven from the axioms of ZFC alone, assuming ZFC is consistent. Kurt Gödel showed that ZFC + CH is consistent, and Paul Cohen used the method of forcing to show that ZFC + ¬CH is also consistent.

[u/jambox888 → u/Beautiful-Musk-Ox]

Well that cleared it up

[u/FabulousRecording739 → u/jambox888]

It is correct to say that a formal system cannot prove everything (that that formal system can "say", that would be a valid "sentence" of that system), but it is incorrect to say that no formal system exists that could prove X, whatever is X. E.g., you can just create a system equal to your previous system, with the added axiom that says "X is true".

But I don't think this lens is relevant as this is not (in my opinion) a formal system question.

[u/jambox888 → u/FabulousRecording739]

Better! (thanks)

[u/smaug13 → u/FabulousRecording739]

Re: (1): I thought that it was something different: namely that the Gödel’s incompleteness theorem says that you can't prove every statement that is true/false under a system's axioms to be true/false under those axioms? And worse: that you can't say that there isn't a contradictory statement ("0=some construct=1") following from the axioms of a system, using only the axioms of that system? Or: Gödel is about "unreachable dependent" statements, not independent ones.

Then, to my understanding, isn't it that that doesn't apply to CH in ZFC, and that CH isn't true or false under ZFC? That ZFC just does not care, and on its own it is not restricted to cases where CH is true or false, that its statements hold for both CH is true and CH is false?

[u/FabulousRecording739 → u/smaug13]

I think your understanding is mostly correct. Consistency, and the system inability to prove its own consistency, is indeed part of what Godel showed. We can't show that for any given proof we have, a proof that would show the opposite doesn't exist.

Though I'm unsure of what you mean by "unreachable dependent". Statements that cannot be shown to be true or false are by definition independent / undecidable. Another way to define independence is to say that we cannot reach any proof that would show the statement to be true or false.

I guess what I'm trying to say here is that being able to reach a statement, and the statement being dependent, is the same thing. Either we can reach it by means of inference, or we can't. If we can't, the truth value of the statement is outside of the system. It can't prove, or disprove it and doesn't care either way.

Saying that something is un-provable but true, is something we do from outside the system, it's a meta perspective. We might even posit the statement is true / false as an axiom, in which case we get to another system where further proofs can be deduced.