# Implications of an infinite versus a finite universe

post by Kawoomba · 2012-12-21T17:12:59.979Z · score: 3 (12 votes) · LW · GW · Legacy · 45 commentsHi gang,

for the last several months, I've intermittently been wondering about a curious fact I learned.

You see, I was under the impression that the universe (as opposed to just the currently observable universe or our Hubble volume) must be finite in its spa[t|c]ial dimensions. I figured that starting with a finite area of space, expanding with a finite (even if accelerating) expansion rate, could only yield a finitely sized volume of space (from any reference frame), a fraction of which constitutes our little Hubble bubble.

Turns out - honi soit qui mal y pense - my layman's understanding was wrong: "This [WMAP data] suggests that the Universe is infinite in extent (...)"

Now, most (non-computer-scientist) people I've bothered with that answered along the lines of "well, it's really big alright? (geez)".

However, going from any *finite* amount of matter/energy to an actual *infinite* amount (even when looking at just e.g. baryonic matter from the infinite amount of galaxies) still seems like a game-changer for all sorts of contemplations:

For example, any event with any non-zero probability of happening, no matter how large the negative exponent, would be assured of *actually* happening an infinite amount of times somewhere in the our very own universe (follows straightforwardly from induction over the law of large numbers).

Such as a planet turning into a giant petunia for a moment, before turning back.

The universe being infinite doesn't make that event any more likely in our observable universe, of course, but would the knowledge that given our laws of physics, there is an infinite amount of Hubble spaces governed by any sorts of "weird" occurrences - e.g. ruled by your evil twin brother - trouble you? Do we need to qualify "there probably is no Christian-type/FSM god" with "... in our Hubble volume. Elsewhere, yes."?

The difference, if you allow me a final rephrase, would be in going from a MWI-style "there may be another version - if the MWI interpretation is correct - that I cannot causally interact with" to a "in our *own* universe, just separated by space, there is an infinite amount of *actual planets* turning into *actual petuniae* (albeit all of which I also cannot interact with)".

## 45 comments

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"This WMAP data suggests that the Universe is infinite in extent (...)" in actuality means "as far as we know, the Universe is spatially flat, and we did not detect anything resembling spatial periodicity or an edge (domain wall)), at least not yet". Don't go all Boltzmann Brains over one careless statement.

You're the phycisist (IIRC), isn't the currently favored model that there is an actual infinite number of galaxies? If so, that's what we should base our discussions on.

There are no infinities in physics, though there are many in the mathematical models it uses. The latter is because infinities actually make many concepts and calculations simpler.

There are no infinities in physics, though there are many in the mathematical models it uses.

This confuses me. What do you mean by "physics" besides mathematical models? You're not lapsing back to realism, are you? :)

Right, I asked myself the same question when writing my comment. I mean that infinities cannot be measured and that the current observations can be explained with infinity-free models, though sometimes not as elegantly.

There are no *intensive* infinities in physics.
That roughly means you cannot have an infinite amount of something in a finite volume of space. It doesn't stop you you having an infinite amount of space.

Which kind of model does an instrumentalist use to guide their actions in this situation? The ones with infinities, the ones without, or a probabilistic mixture of the two kinds? It seems like you're saying just use the ones without infinities (since in either of the other two cases we do have to deal with Boltzmann brains). But how do you justify that?

Which kind of model does an instrumentalist use to guide their actions in this situation?

In what situation? Calculus of infinitesimals is a convenient tool invented by Newton and Leibnitz to calculate planetary orbits and other things. The same can be done with differences and not differentials (and is, for numerical calculations), but requires more work. The situation is similar in most other areas. Kronecker delta is a convenient tool in physics and electrical engineering, without necessarily meaning that there is an infinitely strong and infinitely short spike of current somewhere in your circuit.

In the situation that infinite and finite cosmological models can both explain all current observations. Isn't that the topic of this thread?

The original concern was that the evidence points to the infinite universe, therefore everything imaginable happens somewhere. While this can be a fun speculation, I am merely pointing out that the evidence so far points to a very large universe, but not necessarily infinite or even close to large enough for Boltzmann brains.

That's what I argued on another site a few months ago, but how do you square that with most current reputable sources saying the universe is actually infinite?

This paper gives a lower bound of 251 Hubble spheres, but also leans towards ", the spatial extent of the Universe is inﬁnite" (if the curvature is >=0). Nothing about "just a theoretical construct". As theoretical as all the other laws and the practical inferences we draw from them.

This paper says outright **"An inﬁnite universe is compatible with the data at a conﬁdence level of 4.3σ."**

Those were the two that came up at first glance on Google Scholar when looking for "size of the universe", they weren't cherry picked, and seemed to be follow-ups from the WMAP data.

From the last paper you linked:

There is an indication that a finite universe fits the data better than an infinite one. However, the “standard” 5- criterion for a discovery, corresponding to a confidence level = 5.7 × 10−7, includes the value L = ∞.

The first paper only estimates the maximum curvature, not the size.

Hmm hmm, thanks. If it turns out that our current physical models imply that the universe is in fact finite, my "concerns" would go poof. You seem to be leaning towards finiteness, if so, may I ask what you base that on?

See my replies to Wei Dai.

I feel like this is of a kind with the recent MWI thread :D What's so great about "really existing, with density epsilon" that's so much better than "happens with probability epsilon?" This may just be a case of needing to indoctrinate people in decision-fu a bit more.

What's so great about "really existing, with density epsilon" that's so much better than "happens with probability epsilon?" This may just be a case of needing to indoctrinate people in decision-fu a bit more.

Not sure what you mean by "decision-fu", but there's nothing in current decision theories that says you have to treat these the same. See also this post of mine which may be relevant.

What's so great about "really existing, with density epsilon" that's so much better than "happens with probability epsilon?"

Suppose that I had a plan that will wipe out humanity with probability 0.999999999, but will lead to a positive Singularity with probability 0.000000001 minus epsilon. I understand why that would be a terrible plan (unless the probability that we'll wipe ourselves out anyway is on the same order). Now suppose I had a plan that will wipe out 0.999999999 of humanity's measure, but lead to a positive Singularity in all but epsilon of the remaining measure (and this plan, if executed at all, will be executed by *all* but epsilon versions of humanity throughout the spatially infinite universe). On reflection, after understanding better, would we conclude that this is also a terrible plan? I strongly suspect so, but I can *not* explain exactly *why* it would be. All our experiences would presumably be "experienced much less" in some sense, but what does that mean exactly, and why should I care? The decision-fu I know does not make my confusion go away.

Suppose that I had a plan that will wipe out humanity with probability 0.999999999, but will lead to a positive Singularity with probability 0.000000001 minus epsilon. I understand why that would be a terrible plan

Hm. Okay. Maybe you could explain it to me then. Why is it a terrible plan, even though all the future people who count will think it was great? None of these futures *actually* exist to choose between, you're just planning based on a model of the world. Why not just plan so that the people who matter in your model will have a good time? "When I am, Death is Not, and when Death is, I am Not," so why should we weight it negatively?

(I'd actually be interested in how you'd answer this).

My favorite answer: It's just like how you shouldn't do meth. "Meth" here refers to any drug that rewires your utility function so that you feel really great, right until you drop dead, though you don't care about that part as much when you're on meth. If you just want to make your future self happy by *their own* standards, you should do meth - it feels great once you do it!

Similarly, in the case of either quantum or classical russian roulette, you may be rich if you win, but "zooming in" on only the person who wins is a type of trying to make this model person happy by their own standards. This is a *generally* flawed decision procedure. You should be able to choose based on your own standards, not the standards of the people who live / are on meth.

**[deleted]**· 2012-12-22T14:00:44.045Z · score: 0 (0 votes) · LW · GW

Suppose that I had a plan that will wipe out humanity with probability 0.999999999, but will lead to a positive Singularity with probability 0.000000001 minus epsilon. I understand why that would be a terrible plan

That means that the utility of a positive singularity (on a scale where the status quo is at 0) is less than -10^9 times the utility of wiping out humanity.

Now suppose I had a plan that will wipe out 0.999999999 of humanity's measure, but lead to a positive Singularity in all but epsilon of the remaining measure ... On reflection, after understanding better, would we conclude that this is also a terrible plan? I strongly suspect so, but I can not explain exactly why it would be.

This is left as an exercise. Hint: the previous paragraph of mine may be relevant.

Doesn't need to affect your decisions to have a psychological impact.

EDIT (x2): Imagine we found out we're surrounded by Cthulhu-type monsters on all sides, but luckily we'd be certain they can't causally interact with us. Business as usual, then? Also, wouldn't that imply e.g. that if a relative of yours on a spaceship were going over the cosmological horizon, then "relative = dead, stopped existing" and "relative just forever outside my reach" would be considered identical, because both reduce to the same decision tree? I'd disagree with that, too. Different states of mind can lead to the same actions, yet the difference may matter epiphenomenologically.

**[deleted]**· 2012-12-21T19:21:32.316Z · score: 0 (0 votes) · LW · GW

Doesn't need to affect your decisions to have a psychological impact.

At that point I don't see why you'd call it "implications". What are the implications of a spider getting thrown on your face? You get lots of fear, but it doesn't have implications for ideal morality and rationality besides "don't throw spiders on people's faces".

"implications" has too many implications...

I don't want to hurt your sanity, but we've always been surrounded by Cthulhu-type monsters on all sides, and they can't interact with us. And yeah, business as usual on that one.

And recall, my comparison was not between really existing and being dead - it was between definitely existing with some density, and probably existing with some probability. If your relative definitely exists with a density of 1 in that patch of spacetime over there (unit is "relatives per patch of spacetime you can point to"), then the comparison is to a case where they probably exist with probability 1, which is kinda boring :P

I agree that state of mind is important. But that's *always* important, and thus not very informative :)

**[deleted]**· 2012-12-21T19:15:10.186Z · score: 3 (7 votes) · LW · GW

For example, any event with any non-zero probability of happening, no matter how large the negative exponent, would be assured of actually happening an infinite amount of times somewhere in the our very own universe (follows straightforwardly from induction over the law of large numbers).

No. Probability is in the mind. For example, I am quite sure that the 100th digit of pi is the same everywhere, even though it has a rather wide probability distribution. Likewise for any deterministic system.

Maybe you could say anything with a finite *physical frequency* is happening somewhere, but I think that's just a tautology; How would you discover the physical frequency without looking at the whole universe?

I suppose we could observe from quantum mechanics that there is in fact some amplitude for "planet turns to giant ball of daisys", and conclude that it happens somewhere.

Anyways, I'm with manfred on the decision-fu comment.

For example, any event with any non-zero probability of happening, no matter how large the negative exponent, would be assured of actually happening an infinite amount of times somewhere [...]

No. Probability is in the mind. For example, I am quite sure that the 100th digit of pi is the same everywhere, even though it has a rather wide probability distribution. Likewise for any deterministic system.

Maybe you could say anything with a finite physical frequency is happening somewhere, but I think that's just a tautology; How would you discover the physical frequency without looking at the whole universe?

To ease communication, you should consider "probability" in this context to mean "*expected* physical frequency".

**[deleted]**· 2012-12-23T22:15:51.607Z · score: 3 (3 votes) · LW · GW

Does that even help? The "expected physical frequency" of the 100th digit of pi is the same as the EPF of the 16th digit of the mass of a given object, but my expectation of 100 independent samples of the first is drastically different from the same for the second.

My "in this context" meant in the context of: "For example, any event with any non-zero probability of happening, no matter how large the negative exponent, would be assured of actually happening an infinite amount of times somewhere in the our very own universe"

The term "probability" doesn't *always* refer to expected physical frequencies.

**[deleted]**· 2012-12-23T23:53:57.894Z · score: 1 (1 votes) · LW · GW

(Edited)

Expected physical frequency always matches probability. This specific context has the same problem. If I believe that superintelligence is possible at 50% confidence, it does not follow that on 50% of candidate worlds, superintelligence develops, and on the others, it doesn't.

Well, of course it doesn't necessarily follow. The fact that a resource-limited agent's estimates sometimes prove inaccurate doesn't imply that their estimates are not probabilities. E.g. see here:

Probability is a measure of the expectation that an event will occur or a statement is true. Probabilities are given a value between 0 (will not occur) and 1 (will occur). The higher the probability of an event, the more certain we are that the event will occur.

**[deleted]**· 2012-12-24T01:21:45.367Z · score: 0 (0 votes) · LW · GW

Yes sorry, I fucked up the wording/thinking there. EPF always matches probability, and estimates are always probabilities.

Edited the parent to have clearer wording.

I suppose we could observe from quantum mechanics that there is in fact some amplitude for "planet turns to giant ball of daisys", and conclude that it happens somewhere.

And going from there, that it happens infinitely many times. I'm fine with talking about physical frequencies instead, the actual frequency is irrelevant for the purposes of this topic as long as it's non-zero, which we can conclude from quantum mechanics.

So do you then actually believe there are infinite planet-sized balls of daisies out there, and if so, do you really not care in the least?

I think the point, if there is a point, is that you will almost certainly never causally interact with that planet-sized ball of daisies, so it is a waste of resources to spend too much time thinking about it.

Another way of saying it is this. Case A is that the universe is finite and that cows spontaneously turn into chickens with probability epsilon. Case B is that the universe is infinite and epsilon of all cows actually turn into chickens. In both Case A and Case B your experiences are exactly the same. Your expectation of observing a cow turning into a chicken is the same.

This ties into the long-running debate over waterfall ethics which I think of as important from an FAI perspective. When you're thinking about the moral value of the notional torture of humans simulated by the encoded movements of sea snails on a rock, at some point you have to ask yourself not, "Does this 'torture' have moral value?" but "Am I sufficiently causally entangled with this 'torture' that it has moral relevance to me?"

Case A is that the universe is finite and that cows spontaneously turn into chickens with probability epsilon. Case B is that the universe is infinite and epsilon of all cows actually turn into chickens. In both Case A and Case B your experiences are exactly the same. Your expectation of observing a cow turning into a chicken is the same.

Not quite true... This only follows in Case B if you assume a principle of mediocrity i.e. that you are a "typical" observer. But that assumption leads to known problems in an infinite universe e.g. it implies a strong form of Doomsday argument, where only a tiny fraction of civilisations survive and become space-colonizing. (Otherwise, you'd expect to be part of a long-lived, space-colonizing civilisation of huge population, rather than still on the planet of origin of your civilisation with only a few billion population).

If you DON'T assume a principle of mediocrity, then Case B makes extremely weak predictions about what you should observe : you might, for all you know, be part of the epsilon who observe weird things at some point in their lives.

But that assumption leads to known problems in an infinite universe e.g. it implies a strong form of Doomsday argument, where only a tiny fraction of civilisations survive and become space-colonizing.

What is the problem with this argument, besides an unpleasant conclusion? We haven't actually *seen* any space-colonizing civilizations, and we would expect to see at least some unless we were either one of the first civilizations in our light cone or if most civilizations weren't spacefaring (possibly because they don't survive that long).

Well the main problem is the sheer severity of the Doomsday effect.

Suppose space-colonizing civilizations have an average population a billion times that of "doomed" civilizations, then the mediocrity argument implies that fewer than 1 in a billion civilizations become space-colonizing. If the population ratio is a trillion, then fewer than 1 in a trillion become space-colonizing.

But there are something like 10^22 stars in the observable universe, and a space-colonizing civilization could reach a very large portion of them; further, it would tend to do so, if there is no real competition from other colonizing civilizations (the competition would instead be arising at the edge of the expansion wave, causing travel speeds to increase and approach the speed of light). So the most likely population increase factor is something like a billion trillion or more, implying a chance of civilization survival of 1 in a billion trillion or less. That does seem unreasonably pessimistic.

Fair enough, and on reflection I agree that those kind of survival odds are unreasonably pessimistic given the information we currently have.

"I don't care about whatever cannot causally interact with me", that's the theme of the answers so far. Yet, we are invested in deciding among unfalsifiable propositions in many issues. E.g., even if there may never be an experiment to tell apart MWI from some non-MWI variants, if you'll never causally be influenced by that difference, plenty of passions and words are spent on that. Knowledge in itself can be a terminal value, and it is for many, even if that knowledge has no utility in so far as it does not influence your actions.

If you know there is an actual (insert your favorite SciFi franchise)-themed Hubble volume out there, and *all that separates from you is a lot of space*, would that not be worth contemplating?

Incidentally, do you believe that?

I was under the impression that the universe might be infinite in space without having an infinite amount of matter and energy as in the vacuum is infinite but there is a fixed amount of matter/energy.

**[deleted]**· 2012-12-22T13:56:07.062Z · score: 0 (0 votes) · LW · GW

What is it to you if a planet turns into a petunia, if it can't possibly affect you and you can't possibly affect it?

(a) The Christian god is supposed to be the Creator of the Universe, so the size of the universe has no effect on the probability of his existence.

(b) God is supposed to be omnipotent (thus having the power to defy all physical laws) and omniscient (having every bit of information).

I don't think the size of the Universe will make such a god any more improbable.

**[deleted]**· 2012-12-21T17:25:17.961Z · score: -6 (6 votes) · LW · GW

Well, the universe having a finite amount of space (not just a finite amount of energy/matter) would imply some sort of unbreakable, impassible "edge of the universe", which seems pretty ridiculous to me. Now in an infinitely large universe, finite energy/matter will continue to expand asymptotically towards 0 energy/matter per unit space. If there is infinite energy/matter, it converges to a finite density (assuming the spacial infinity and energetic infinity are in a linear ration).

No, not necessarily. If, for example, it could be represented in higher dimensions as a hypertorus or a hypersphere, it would be finite in extent yet not have an edge.

I'm not questioning why the universe would be infinite, rather than finite.

Also, most sources I can find strongly hint at there also being an actual infinite number of galaxies, Max Tegmark for example. It's not a fringe idea certainly. If I find a single authoritative and concise source, I'll link it. This comes close.

This'll probably get down voted, but:

Dude... NO EDGE

Sorry, but "this'll probably get down voted, but" just doesn't work here.

I figured.

Just wanted to imply that I wasn't trolling to anyone who doesn't get Vlogbrothers references.