I'm downvoting primarily to discourage deliberately sensationalist titles. I don't want to start seeing "What this AI Gatekeeper did will shock you!" and "Five reasons why MWI will show you how everything you thought you knew about quantum mechanics is a lie!" and "These ten effective altruists will restore your faith in humanity!"
Aside from the sensationalism, I remark that you're (re)defining "simulation" to mean "world whose behaviour is equivalent to that of a simulation" and that the axiom you adopt in step 1 already transparently implies that our world is such a world. So you kinda do have a proof -- but it's of the form "Suppose p. Then it follows that p.".
None of the replies to this post actually address the crucial part of the argument: the step from the existence of mathematical truths beyond the reach of a specific formalization of mathematical reasoning, to the existence of a reality beyond or outside a specific physical theory of everything, on the grounds that a mathematical theory of physics must be subject to Goedelian incompleteness.
Step 5: The consistency of our Multiverse can be proven in a formal system which describes physical laws of a larger world, in which our Multiverse is a small part of, essentially a simulation.
Are you assuming that every possible multiverse exists? If so, it seems pretty obvious that there's a universe that contains ours. If not, you still have to prove that the multiverse given here exists.
Step 3: By Godel's incompleteness, there are some theorems in this formal system that cannot be proven.
No. That is not what Godel's incompleteness theorem says.
Given the extremely negative reaction, I will delete this post tomorrow, unless someone offers a good reason not to.
It may serve as a helpful example to other users of what sorts of posts receive negative feedback.
I suspect based on how my comment is voted up that much of the downvoting isn't the content so much as the delivery.
I also think that is the cause of almost all the downvotes, the numbers match well. The ideas presented are about interesting and relevant topics so potentially worthwhile even if the specific proof does not hold up. If not for the appearance of certainty (esp. "mathematical proof") where you should have been offering it up as something you'd like thoughts on (esp. if you had specific questions, pointed out parts you were less sure about, etc) I would likely have upvoted, as is I only didn't downvote because it seems you've lost more than enough karma for this already and it would feel like kicking a dead horse. I'm sure by now anyone who sees this will realize clickbait titles are not a great way to get points on lesswrong.
I expected at least one reply of the form "your argument sucks, but let me try to steelman this one part of it..." but alas.
Because it is already downvoted to invis for anyone who cares, and some of us found it mildly amusing.
Quick search of "define simulation" yields this:
Simulation is the imitation of the operation of a real-world process or system over time.
You say ( I paraphrase): We live in an unbreakable simulation and can't break out, but so does every level above us. And presumably every level below us as well.
So I might ask, a simulation of what? A simulation of some reality that is not a simulation? But I thought you just said every level is an unbreakable simulation.
So I might ask, what is a simulation, or what do you mean by a simulation? Because in my thinking, if you're simulation is not a simulation of something else, then it is reality. So we live in a universe that has physical laws, which means it goes clunking along according to rules we did not write and cannot change. To you that sounds like a simulation, but to me, in the absence of it being a simulation of something else, it sounds like reality.
Right, a good question. I did not mean a simulation of something, rather "let's set up some rules and see what happens". In the same sense that the Conway's Game of Life is a simulation of life. And yes, to those inside it is reality. And to those outside it is a small subset of reality. And my original point was that those inside can potentially infer that there must be something outside by hitting the limit of what they can explain. But they have no hope of breaking out. This last argument, of course, relies on the tenuous assumption of the physical laws being mathematical structures and every event in the world being, in essence, a theorem.
Cool enough.
Am I correct that you suggest an infinite regress, or rather progress? That every higher level is also a rule-following game set up as a subset of a (presumably) yet more complex level above it? And that there is no end to this progression, no top level?
I did not mean to suggest it when I started writing the original post, but the logic, flawed as it is, seems to be pointing that way.
You can neither prove nor disprove that we are in a simulation, just as you can neither prove nor disprove the existence of God. In both cases, you cannot come up with a measurable experiment, your claim is not falsifiable, so you cannot use the scientific method to prove or disprove it, and the discussion will inevitably lead to an argument about semantics: "how do you define simulation", "how do you define god", "how do you define define", etc.
Almost all similar arguments I witnessed ended in an argument about semantics, where one came up with his own definition of a term and tried to prove or disprove it, instead of focusing on a commonly accepted definitions agreed by both parties.
Present an argument as a proof, or an outline as a proof, and mathematicians will take that as an invitation to tear it to shreds.
I hear them growling now.
Actually, the title is a sensationalist lie designed to attract attention. I have no proof. Obviously. I'm not a mathematician. But if I did, it would go something like the following.
Step 1: Assume that there are ultimate laws of physics governing everything in the world. Say, the wave function of the Universe, whose knowledge allows one to know the Multiverse, as it was, is or will be. Or some other set of laws.
Step 2: Write these laws as a mathematically consistent formal system representing something akin to the Tegmark Level IV Ultimate ensemble.
Step 3: By Godel's incompleteness, there are some theorems in this formal system that cannot be proven.
Step 4: By construction, these theorems correspond to physical laws whose origins must forever remain a mystery to those inside the Multiverse, because they are a part of it.
Step 5: The consistency of our Multiverse can be proven in a formal system which describes physical laws of a larger world, in which our Multiverse is a small part of, essentially a simulation.
Step 6: Since we cannot determine the origins of our own physics, we cannot figure out a way to break out of our simulation.
On the bright side, there is a Corollary: Every level above us is also a simulation, so we are not alone!