## Densities in Infinite Sets (Simulation Argument).

• 99
If you're not familiar with Bostrom's simulation argument, I'll briefly summarize (badly, I'm sure):

If extremely granular simulations are possible, and beings are interested in making them, and it doesn't require tremendous resources, it's plausible that there exists two sets: those who are in a simulation (set A) and the those who create simulations (set B). The population of set (A) would most likely vastly outnumber (B). If you don't know if you're in a simulation or not, the odds are you're in a simulation.

OK, that argument works for a finite universe, but not in an infinite universe. In an infinite universe, sets (A) and (B) are both infinite and infinite countable sets are equal. The argument can't work if both sets are equal (or at least, if can work, but it generates a weaker conclusion: if you don't know you're in a simulation, there's a 50/50 chance you're in one).

Here's an attempt to get around the infinite universe problem:
To deal with these infinite cases, we need to do something like thinking in terms of densities rather than total populations.A suitable density-measure can be finite even if the total population is infinite. It is important to note that we to use some kind of density-measure of observation types quite independently of the simulation argument. In a “Big World” cosmology, all possible human observations are in fact made by somebody somewhere. (Our world is may well be a big world, so this is not a farfetched possibility.). To be able to derive any observational consequences from our scientific theories in a Big World, we need to be able to say that certain types of observations are more typical than others. (See my paper “Self-Locating Belief in Big Worlds” for more details on this.)

The most straightforward way of making this notion precise in an infinite universe is via the idea of limit density. Start by picking an arbitrary spacetime point. Then consider a hypersphere centered on that point with radius R. Let f(A) be the fraction of all observations that are of kind A that takes place within this hypersphere. Then expand the sphere. Let the typicality of type-A observations be the limit of f(A) as R--->infinity.

To apply this idea to the case where we might be living in a simulation, we can use a similar rule, except that we define the seed point to be the location at the bottom level of reality where the computer is located that is ultimately running our simulation, and we take the hypersphere expansion at that level of reality. (One could modify this rule so that it would work also in the case where it is possible that there is no lowest level of reality but rather an infinite regress of simulations within simulations “all the way down”.)

Re the bolded part: how can you do a density measure of part of an infinite set?
• 287

Before getting to the density of the infinite sets, l would like to comment on the following remark
sets (A) and (B) are both infinite and infinite countable sets are equal. The argument can't work if both sets are equal
Consider the people in simulation to be set of natural numbers and the stimulators to be set of even numbers, they will have the same cardinality but they won't be equal.
The most straightforward way of making this notion precise in an infinite universe is via the idea of limit density. Start by picking an arbitrary spacetime point. Then consider a hypersphere centered on that point with radius R. Let f(A) be the fraction of all observations that are of kind A that takes place within this hypersphere. Then expand the sphere. Let the typicality of type-A observations be the limit of f(A) as R--->infinity.
I believe the best way to compare two infinite sets of same cardinality is by density measure, and the density of natural numbers will will greater as R ---> 100 such that f(a)=100 and as R--->100, even sets will have f(a)=50.

But yes, if we extend R to infinity, how will that work to compare densities. How would you define density, is density equal to rate of growth or observation ?
Really interesting but confusing too.
:smile:
• 99
I thought it was a rule that if infinite sets are countable, they're equal. No exceptions.

Informative post!
• 425
What does density mean at that point? Cardinality is the number of members of a set. What else can there be to a set besides that??
• 1.1k
I thought it was a rule that if infinite sets are countable, they're equal. No exceptions.

The even natural numbers and the odd natural numbers are both countable but they're not equal as sets. They're cardinally equivalent, but that's not the same thing as set equality. In set theory two sets are equal if and only if they have the same elements. That's the axiom of extensionality. https://en.wikipedia.org/wiki/Axiom_of_extensionality

In any event simulation theory is essentially a theological argument. If I say God makes every blade of grass, made me the way I am (so THAT explains it!), is omnipresent, omniscient, and omnipotent, you'd call me a religious true believer or a religious nut, depending on your manners and theological beliefs. If I said the great computer in the sky did it, you'd invite me to give a TED talk. There's no difference in the argument. Whether I'm a program running in the great computer or a thought in the mind of God, what is the difference? Both claims are strictly outside of science. They're not subject to observation or experiment. What kind of experiment could distinguish the two, and distinguish either of them from naturalism, the idea that the stuff out there is real and we live in a physical universe that is explainable by natural processes?

I believe the best way to compare two infinite sets of same cardinality is by density measure,

Natural density is useful in some contexts, but it's not countably additive hence is not, strictly speaking, a measure or a probability measure. That's its drawback. It does have its uses, but to claim it's the best way to compare countably infinite sets is asking it to do too much. There's no uniform probability measure on a countably infinite set, and that's a fact we all have to live with.

https://en.wikipedia.org/wiki/Natural_density
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