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Emlyn O'Regan wrote:

I like your argument, Rafal. If I might summarise, I think this is what you

were saying...

- There are universes N which occur naturally, from "pure mathematics",

which contain humanlike civilisations (enough like our universe for purposes

of this argument).

- There are universes S which occur as simulations inside other universes,

which are humanlike civilisations (as above).

- The space of all mathematics of which N is the same order is larger than

the space of simulations S, thus we are more likely to be in N than S, and

thus not in a simulation.

Is that about right?

### It is a very good summary.

---- If so, I would counter by saying that the space N cannot be larger than the space S, and in fact should be of a lower order (aleph thingy?) than S. Take this from the hierarchical conception of the full space of applicable universes. Note that universes in N spring out of "mathematics per se". Whereas elements of S can derive from elements of N, or from other elements of S.### That's right. But I would think that the number of repetitions of our visible universe (V), derived directly from any branch of N, is larger than the number of instances of V produced within that branch by way of S. I could write it as :(N->V) >> (N->S->V). See below. ---- So, if each n in N has some non-zero probability of creating a sub-hierarchy of elements of S, where the depth of that hierarchy is infinite, we get an infinite number s's in S for each n. ie: the order of S is higher than N, so that you are vanishingly unlikely to actually exist in an n, rather than an s.

### Here I would disagree - the depth of a simulation hierarchy in S is not infinite. I am afraid my powers of abstraction and intuition are faltering at these dizzying heights, but by analogy to cellular automata, I would claim that any simulation is much simpler than the device implementing it (you could say, "The thought is always simpler than the mind"). The number of elements within a simulation is always smaller than the number of elements in the surrounding non-simulated universe.

---

Now, I am not sure if your argument shows below that the order of N is as high as it can be (equating it with the space of all mathematics?). I'd question that claim, if you are making it (not sure). If that was the case, it would follow, I guess, that the space of simulations must be constrained in some way to match it.

### I cautiously surmise that the order of N is higher than S. If the cellular automata analogy holds, then it would provide the constraint on the order of S.

We could try to think about it in a more physical way. If you imagine a physical universe with an infinity of domains of expansion (Big Bangs), then an infinitely large subset of these domains would contain exact copies of ourselves (but it would be an infinity of a lower order, because only a small fraction of the domains would be indistinguishable from ours, ). If you consider a finite sample of this universe at any point in time t1 (the imaginary time of Hawking), there would be a finite number (n(t1)) of domains of the type V (visible universe indistinguishable form ours). If all of these domains further evolved to contain multiple ancestor simulations (s), (corresponding to billions (B) of posthuman civilizations running separate but identical simulations), then at time t2 the total number of simulations would be:

s(t2)=n(t1) x B

At the same time, if the rate of spontaneous generation of new V domains in that sample of universe was E (=explosion), the total number of V type domains at t2 would be:

n(t2)=n(t1) x E

This is saying that for any V domain present at t1 there would be a constant number of new, non-simulated, physical domains generated until t2.

If E>>B, then at any point in time n>>s, and for the whole infinite universe N>>S. Of course we have at present no way of estimating either B or E in our universe, and even if we did, there might be branches of mathematics predicting (or rather directly generating) different B and E.

This makes me doubt that we can rigorously calculate the probability of being simulations.

Aren't there some cosmologists advocating the infinitely bubbling universe?

Rafal

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