Well, reality is what we perceive. A computer can only simulate worlds which are less complex than the world in which the computer exists, but if the simulation is closed, its inhabitants have no way of proving that it's a simulation. They simply have no way of knowing how things are in the real world. Even bugs in the simulation would appear as an empirically found law of physics to them. A laser in such a world would not exist except as a function of the basic elements that exist in the simulation. However, such a simulation would obviously need to either be seeded without science and develop it by itself or overthrow the science it was seeded with.

Surely the results of such experiments could be faked. For example, it could be simple to build a 'matrix' where the value of PI could be, say, 5, or where Newton was correct and light permeated space the instant it was emitted and mass had not effect on time.

Hell, there are limits to our own understanding of both the extremely small and the extremely large. What if those limits are not that far from the limits of our "simulation"? How would you tell? Build bigger accelerators/telescopes? How big would they need to be?

Our knowledge of "what should be" is based purely on obseravtion. We're always testing the boundaries of our knowledge. But who's to say that when we delve deeper into the depths of the cosmos we won't discover a message:

"game over, insert coins to play again."

or

"Hi, this is God, I'm not in right now, please leave a message."

Incorrect. All that needs to be simulated is what you actually perceive. In modern games, the engine calculates what can and can't be seen and doesn't draw the things that can't be seen. A simulation would use a much more sophisticated version of that algorithm. If you're looking through a microscope, microbes are individual simulated. If you aren't looking through the microscope, then they aren't simulated, or are simulated in the aggregate to calculate gross effects that might be perceivable (such as tainted meat causing food poisoning.)

Remember, the simulation has to know exactly what you're doing and what you're perceiving in order to feed the information to your brain. If you turn your head, that isn't a physical motion. The simulation detects the impulses that indicate you desire to turn your head, and adjusts your visual and physical feeds to simulate that motion. So it's certainly capable of determining that you are peering through a microscope and adjusting the level of detail accordingly. How detailed is the simulation? Precisely as detailed as it needs to be, but no more.

One interesting result of this is that observation would affect the behavior of the universe. Also, changes in the environment, such as the presence of a second slit in a screen, might alter the algorithm used to calculate the behavior of, oh, I don't know, maybe photons.

Actually we do know the Universe's smallest "pixel size": the Plank Scale. Who's to say whether this is a computational limit imposed upon our simulation by external beings or a true physical limit?

Also, it isn't actually true that a computer cannot simulate soomething more complex than itself. If time is no object, it can simulate something a million times more complex than itself in a very long period of time. Who's to say that maybe a single second in our simulated world takes a million, maybe even a billion years to compute in "real time"?

I think a lot of people are missing a key points.

Godel's theorem in a nut shell: you cant prove inconsistency in any set of axioms within the context of those axioms.

suppose for a moment that this is a simulation with a finite amount of memory to parameterize the "world". the state of this system is propgated from time slice to time slice by some set of finite difference equations. well this means that everything is perfectly self-consistent. if you devise any experiment within the simulation itself to measure any observable then you will discover it is self consistent. The laws of nature a person living there would formulate would in fact be the correct ones for that system. you would never be able to discover an inconsistency.

consider for example QM. basically in a quantum world there ARE limits on resolution. indeed the limits are surprisingly like how one creates a simulation. for example, in any practical 3-D game the voxels of distant objects have larger volumes than the close by ones that you can see more clearly. likewise fast moving objects in the background are less precisely placed from frame to frame while maintaining on average an accurate speed.

its as though someone gridded the game in such a way as to have hyper cubes of constant delta-P time delta-X. hey wadda ya know that's the heisenberg uncertainty principle.

Indeed its easier to simulate a trajectory if you dont have to do it exactly. simply compute the approximate result with error bars and then any time the result is closely inspected you return a different sample from the approximate distribution. Thus one does not have to memo-ize everthing the game player has looked at carefully, you can recreate it on the fly each time something is inspected at high resolution simply by drawing an approximate sample from the distribution. The fact that two looks never quite agree is written off as the "hiesenberg uncertainty principle", or to the QM notion that inspecting an object can change its state.

Another hiesenberg principle is the energy-time uncertaintly (to measure the energy of something precisely takes increasing amounts of time). Again this is in keeping with a simulation. to compute the simulation to increacing levels of precision will take more time.

and remember folks the simulation does not have to run in real time!

Finally to digress a bit. Just suppose for moment the supposition that this is simulation is true. then might it might also be possible that the people doing the simulation are also simulations. and so on ad infinitum. the interesting thing is that at each layer of this onion it seems to me that the plausibility that you live in a simulation increases. this is because with each subsequent layer the plausibility of sufficient computer power prior to extinction improves.

High level emulation. If there is a microscope for you to look through, it is being emulated, then whatever has created the microscope can program it to rewrite everything you look at with it in a way that makes sense to your species.

it would be mind-numbing to write (much less RUN) a program that would fully emulate every atom in the world at all times. all you have to do (ask anyone in movies) is emulate the minumum amount to look realistic on screen. if someone needs to look closer, emulate what they're examining properly, only while they are examining it. Otherwise you can very easily emulate a white box with bumpmaps, rather than the wood, the drywall, the paint, the electricity, and everything else that makes a wall. until someone examines the wall, you can get away with just a white box with paint-like bumpmapping.

Then you simulate what would be seen. Everything could be treated as a surface with a varying transparency and a texture mapped on top of it. You wouldn't have to visually simulate anything smaller than the eye could resolve, but if needed, the simulation could simulate portions in more detail.

It would be easier from a programming standpoint to simulate all of the individual atoms, but that would be prohibitively slow. We're talking tens of thousands of years for less than a second of simulation time using conventional computers on anything less than a planetary scale.

Quantum computers and chemical computers could speed it up greatly, but it would still take massive amounts of raw processing power to keep track of all of those atoms, let alone let anything interact with them.

You can never see anything smaller than the smallest dot that your eye can perceive. However, you can design devices to enlarge objects (or increase the resolution of your eye, depending on how you look at it).

One of the huge problems with The Matrix is the question of how people were actually put into it. If anyone had memories of the real world, then they would undoubtedly find a way to pass them on to their children. So, that implies that none of the first generation of Matrix denizens was ever outside the Matrix at any prior point in their lives. Yet they had parents. The programs in the Matrix aren't compassionate at all, so they certainly couldn't have raised the children. Perhaps they had been imprisoned for millennia, but if that were the case, I would have expected the robots to have wiped out the last of the independent humans. Due to the way memories are stored, there is no way to erase specific memories from the human mind without some serious brain damage. We can only stop new ones from forming. Perhaps the robots were able to create synthetic sets of memories for the first parents, but again, how? That would require someone in the Matrix in the first place so that his memories could be copied. Perhaps the first parents were willing subjects? I don't really see that as in The Animatrix, the general populace was destroying the robots in the streets. That would be like southern whites agreeing to be slaves to some blacks during the Civil War. Very few would. Perhaps enough did that they were the first generation.