Can you explain further?
Maybe I'm too out of this scope but if you want to simulate Universe X plus the computer Y that simulates X then you'd need at least 1 extra bit of memory (likely way more) to encompass the simulation plus the computation running the simulation (X+Y). The computer running the simulation by definition is not part of the simulation, so how can it be that it can truly simulate itself?
In Men in Black II (2002), Will Smith learned that a miniature civilization existed in a locker. Later, Will Smith’s character learned that our civilization was in a locker of a larger civilization.
By thinking of memory usage, you’re restricting yourself to our perceived physical limits withine our perceived reality.
But, what if the things running the simulation did not have those limits? E.g. maybe data could be stored in an infinite number of multiverses outside of the infinite simulations being discussed. Any of the simulations could potentially simulate universes like ours while still allowing those simulations to contain others, to be contained by others, to have references to others, to have reflective references, etc. The makes anything and everything possible while not necessarily removing limits we have in our own simulation. It just depends on what’s running the simulation.
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Not quite, compression enables you to simulate / represent / encode x data with less than x memory.
Only for those inputs that are compressible.
If a compressor can compress every input of length N bits into fewer than N bits, then at least 2 of the 2^N possible inputs have the same output. Thus there cannot exist a universal compressor.
Modify as desired for fractional bits. The essential argument is the same.
Would the compressibility of the state of the universe be useful to prove whether we are in a simulation already? (i.e. it is hard to compress data that is already compressed)
Roughly speaking, Gödel encoded (or “simulated”) the formal part of mathematics within arithmetic (using operations such as addition and multiplication), and constructed a sentence that says “this sentence is unprovable” within that simulation.