I would argue that it is not possible to create a simulation of a world within a world.
I can only imagine that a simulation of our world could be done within a computer?
A computer has a finite memory, however, our world is possibly not finite?
So would a simulation of a world within a computer not be a full replication of our world because of the limitations of memory?
Or is there another method of creating a simulation of our world?
At face-value, this is demonstrably possible. A video game is a simulation of a world. Some video-games have mini games inside them, a simulation inside a simulation. A movie, in some sense, is a simulation of a world, and many movies contain other movies inside them. I assume these examples wouldn't qualify in your view, but that raises the question "what counts as a simulation?" for your purposes.
I'm going to assume you mean something that could convincingly replace the world as we experience it. (Let's avoid the entire question of consciousness by assuming a brain-in-a-vat setup. The brain is conscious, but all its sensory data is generated by a computer.) Could a simulation of that caliber contain another simulation of that caliber wholly inside itself, where the child simulation must be built entirely from entities of the parent simulation (we can't cheat and just simulate both worlds at the root level)? In principle, there doesn't seem to be anything that would prevent this in terms of overall mechanics. Just as an emulation of any computer can be built on any other computer, it should be possible to build one simulation inside another.
However, your question specifically calls out the question of size. Can a universe of the same size/complexity as the parent universe be built inside of it? Intuitively, the answer is no, the child universe would need to be smaller (simpler). However, there are ways around this. There's an old stage trick where the painted backdrop makes the actors look as if they are in the wide-open outdoors, when in fact they are in a small enclosed room. Similarly, many video games use a "skybox" to make the in-game universe seem big and expansive, even though only a portion of it has actually been constructed. Any given person has personally experienced only an infinitesimal portion of our seemingly infinite universe. If it were simulated, it's possible it could be far smaller than we think.
Depends on what you mean.
It is not possible to generate a full resolution simulation of an entire system within that system. This is intuitively obvious. Imagine the smallest possible thing that you could use to store or compute data with. Your base system has n such things, and you need at least one such thing to represent the existence (much less the behavior) of each of those n things. Therefore, to simulate n things, you need n things, or the entire system you want to simulate.
It is possible to generate a low resolution simulation of a system in that system, and it might be possible to generate a full resolution simulation of part of a system within a system. If you only want to simulate n things, and you have n^100 things, you can at least theoretically simulate n things.
If we are in a simulation, we are almost certainly not in a full resolution simulatiom. Within our reality, there are real physical constraints to creating even a theoretical system that could simulate observed reality. Dodges around this, like "we could simulate only what people are looking at" ignore the difficulty of both tracking what people are looking at and ensuring that there is consistency between those things across time and space. That is, one person observes a cloud, and then goes inside, and then goes back outside. The system, when it "recreates" the cloud, has to put it in the right place based on a bunch of particle physics calculations that are exactly as difficult as simulating the thing all along.
It IS possible that we are in a lower resolution simulation of a vastly complex hyper-reality.
Elon Musk argues that it is most likely that we live in a simulation:
In other words, as I understand the discussion, Elon asserts that there is "base reality". Everything else is a simulation perhaps nested in a simulation in daisy-chains that somehow link back to the base reality.
In that case I map base reality to The Great Mystery. But I have no idea how to calculate the odds or probability that we are living in a simulation and not in base reality. Why? Because I exist with probability p = 1, the certain event, and I have no knowledge of how many nested simulations could be in the chain of simulations between me and The Great Mystery.
I would have to use some uncertain assumptions derived from my perceptions of reality to speculatively calculate the odds of existing in a simulation that is not so-called base reality.
I am fairly certain that each mind or body-mind of a living animal on earth generates the subjective perception of reality as a local simulation, that is, as a model of The Great Mystery arising in the mind. I could be a brain in a vat with electrodes stimulating the neurons such that the brain simulates my experience of a mind in my body and a body in my mind and my body-mind in the context of so-called external reality. So then I am back to The Great Mystery as base reality.
That pretty much depends on how you'd define "simulation" and "world" and to the level of detail that you want to consider.
Chris sunami already gave a very good answer, so probably rather read that instead.
First of all you need to distinguish between a simulation and an emulation. The first is an attempt of a practical recreation of something as close to the real thing as you can get (which leaves a lot of wiggle room in terms of the degree of detail), while an emulation is just meant to replicate an effect. So a simulation is something like a crash test, a re-enactment or if you really want to go to computers a computer simulation in which you idk track the behavior of a physical property by incrementally applying the relevant laws of physics that influence it. Idk so if you drop a ball you might look at it's xyz position and track it through the progression of time while the ball is under the effect of gravity for example.
So if you stick to the thing computers were always made for: games. (Let's be real scientific simulations are also just big children playing around with sophisticated toys...). Then you can distinguish for example between the kind of simulations that recreate a universe with self-consistent rules, so for example all objects in this universe are effected by gravity while the less physically inclined game dev might just recreate the same effect by moving sprites representing objects from the top of the screen to the bottom.
So it's essentially the difference between throwing an apple up into the air and catching it and having a picture of an apple tied to a stick and move that stick up or down.
And ironically the former might be easier than the latter with regards to memory consumption. Think about what it would mean to "emulate" if you time traveled to the medieval period. Just find or recreate a setting that would have fit and be yourself... done. While if you wanted it to look and feel functionally equivalent without actually being the same, you'd end up with something like a D&D group larping in an urban setting creating the world by using tomes worth of rules, regulations and explanations to replicate a realness that isn't real.
On the other hand it's quite simply to entrap you in a very large emulated universe, like idk imagine a standing in a field with crops above you is an blue sky with clouds that you can reach, to fill it a bit you might scatter some trees. Now if you make that landscape large enough and make it so that if you'd leave the area you'd reenter on the other side you could recreate the impression of an infinite universe within finite space. Similarly it's totally possible to do that same thing within that universe and so on.
Sure if you wanted to simulate that in an emulation you'd run out of memory because the simulation would need to make use of the resources of the emulation while still being active. That being said if you just emulate that as well you could also show a loading screen and load a different level once the observer looks at it, so it could even have the same size.
Though when it comes to simulations you could also have people in a future society re-enacting a past society that, re-enacts an even more in the past society. So a simulation in a simulation in a simulation. Does that count? I mean they still happen in the same base reality and as such obey to the same base reality laws yet for all intents and purposes the "world", the experienced reality of those larping that would be different. Though it would still be a self-consisting universe in which (with reasonable caveats) things happen because of real life interactions not because of some narrator or rule book telling you that things happen.
We can physically build the system we want to test and actually do the experiment. However we currently have no capacity to simulate anything at all without extreme simplifying assumptions. This includes even things conventionally thought of as nothing, like microscopic regions of vacuum. This includes both doing the simulation on paper or with the aid of a computer. Simplifying assumptions are known to be false and to generate false conclusions. Simulation works despite this because we carefully match our simplifying assumptions with the kinds of things we're trying to simulate, so that for the questions we want to ask, whether or not the assumption is true turns out to be irrelevant. It is currently a scientific, not an engineering constraint: given arbitrarily long time and arbitrarily great resources, we still have no capacity to simulate anything at all without needing simplifying assumptions. But to simulate anything at all such that you can ask any question you like about it is to simulate without simplifying assumptions. To be able to simulate anything at all without needing simplifying assumptions is to have a perfect model of physical law over the relevant scale. To be able to simulate it with observers inside requires a perfect model of physical law over the relevant scale that includes the observer within the system, an added hurdle.
We do not currently know whether it is ultimately a scientific or a logical constraint. That is, it might turn out to be possible to simulate something, or it might be a priori impossible, depending on the character of yet-unknown laws or meta-laws of physics.
If it turns out to be possible to simulate something, we do not know whether or not there will be physically-imposed constraints which pose unbridgeable engineering challenges even if you give the civilization that's supposed to do the simulation as much resources as it's physically possible to have. If your simulator needs to mass more than a galactic cluster in order to hold the state of anything bigger than a snowflake, for example, you're just out of luck (and you should really have just saved a few billion years of effort and built a snow machine). "Bigger" might not even make sense from this perspective - it may turn out that the only way to really simulate anything at all is to simulate the whole universe that it's a part of all at once.
So an honest statement of the simulation hypothesis(a) really runs:
6 follows from 1-5, but that's a whole lot of absent evidence to base suppositions on. I suspect that if "I have played video games; I imagine that I can imagine being a character from a video game, therefore I could be a character from a video game" wasn't really underwriting the intuition, the simulation hypothesis would be a one-line mention in a footnote about obscure iterations of the brain-in-a-vat scenario, of interest only to the most pointy-headed of philosophers.
(a) Makes no in-principle-testable predictions about doable experiments, therefore not a hypothesis in the scientific sense.
