There is no such thing as a probability of 1/infinity. — fishfry

You do get your knickers in a twist with great rapidity. As you know, that was Michael's terminology. I went along with it for the sake of discussion.

But also, 1/infinity is the proper definition of the infinitesimal as far as I'm concerned. Now you will get on your high horse and object no doubt. But I went along with Michael's terminology largely because I also like that sly implication. It is another way of getting across that the probability ain't actually zero even if it is almost surely zero when it comes to an infinite spatial universe producing replica earths with replica people doing replica things.

When people are talking about an infinitely-sided die, I assume they mean a countable infinity, which has no uniform probability distribution. — fishfry

If you remember, it was you who introduced the confusion. I was trying to sort it out for you by pointing out that those would be the kind "coins" you would need to be flipping...

If there are two states and infinitely many universes they could be 0, 1, 1, 1, 1, 1, ...

If 0 is the "earth" state, there is no other earth. 1 is maybe Mars. So Mars exists infinitely many times but not earth. If there are a trillion states, same argument. SOME state recurs infinitely many times, but not necessarily any particular state. Maybe there's only one earth even though there are infinitely many copies of Mars. It's perfectly possible. — fishfry

This still reads as nonsense to me. Maybe you agree now as you seem to have discovered ergodicity and moved on to a notion of a universe chopped up into sufficiently large but finite regions - the ensemble of microstates picture that I also have been at pains to criticise.

You do get your knickers in a twist with great rapidity. — apokrisis

As I attempted to make clear earlier, I'm ignorant of physics and so I have to take on faith a lot of what's written here. But now and then the discussion wanders into areas I'm familiar with; and more often than not what's said is nonsense. At those moments I do my best to straighten out the math.

As far as your snark, it's not necessary.

As you know, that was Michael's terminology. I went along with it for the sake of discussion. — apokrisis

Yes and I corrected it because it's wrong and confuses the subtle probabilistic issues at stake.

But also, 1/infinity is the proper definition of the infinitesimal as far as I'm concerned. — apokrisis

You are entitled to your private definition. The actual definition of an infinitesimal is a quantity x such that 0 < x < 1/n for every natural number n. The gap between your instincts and how math actually works is significant. I understand in general that you do not believe in standard modern math. You made some remark about Cantor earlier along those lines. Philosophically it's perfectly valid that you have alternative ideas about math and don't accept parts of standard modern math. But physics is most definitely based on standard modern math; and to the extent that your outlook diverges from that, you are introducing confusion into the conversation.

Now you will get on your high horse and object no doubt. — apokrisis

Is that really the best you can do? I'm simply doing my best to explain the viewpoint of standard math, which is the math used in modern physics; and you are arguing from your Peircean viewpoint and claiming that you are entitled to make your own definitions for mathematical terms that already have perfectly clear standard definitions.

But I went along with Michael's terminolog — apokrisis

I'm not sure how that supports any point you might be trying to make. I did actually correct his usage first, and I left your first usage alone simply because he was first and you were simply going along. It wasn't till you mentioned it a second time. If that bothers you, I'll apologize if it makes you happy.

For the record I do understand that you have your own private notation and that you reject the standard notation of modern math, on which physics is based. Is that a fair assessment? Since you admit you have your own definition of an infinitesimal, and you assign meaning to the symbol 1/∞. And that I was a terrible person for calling your private ideas, which you are perfectly well entitled to hold, nonsense. They're not nonsense. They're merely your personal ideas and notation, totally at odds with modern math.

But I went along with Michael's terminology largely because I also like that sly implication. — apokrisis

Ok. You are trying to express the idea of an infinitesimal probability. We all have these intuitions of infinitesimals, as Leibniz did. I truly get that. But since we're doing physics, it's important to make sure we get the math right. Else I wouldn't bother to bring it up.

It is another way of getting across that the probability ain't actually zero even if it is almost surely zero when it comes to an infinite spatial universe producing replica earths with replica people doing replica things. — apokrisis

Yes I understand your philosophical point. But your math is wrong. And we're doing physics. If you're using math metaphorically you should say so up front.

If you remember, it was you who introduced the confusion. — apokrisis

This refers to the infinite die. No that's not true. You introduced it, and the moment you did I realized you didn't have any idea what the conversation is about. The duplicate earth argument depends crucially on there being only a finite set of possible states in any bounded region of space. That's fundamental to the argument. When you brought up an infinite state die, I knew you simply had wandered off into some conversation the rest of us aren't having.

I was trying to sort it out for you by pointing out that those would be the kind "coins" you would need to be flipping... — apokrisis

But that's exactly wrong. You need a finite die with a very large number of faces. One for each admissible state of all the particles in some bounded region of space.

It's ironic that just when you are totally losing track of the thread, you think you're helping me. You've done that before and you were wrong then too.

This still reads as nonsense to me. — apokrisis

I'm perfectly willing to stipulate that much of what I write reads as nonsense to you. You have the same effect on me. Your writing seems extremely learned yet you never make a lick of sense.

Why don't we agree not to interact? I was really surprised earlier that you directly replied to something I said. I don't think our interactions are productive. I find your snark annoying, especially since it generally shows up when your degree of wrongness is at a local maximum.

Maybe you agree now as you seem to have discovered ergodicity and moved on to a notion of a universe chopped up into sufficiently large but finite regions - the ensemble of microstates picture that I also have been at pains to criticise. — apokrisis

As it happens I've spent the afternoon chasing down ergodicity. I do know a little about it relative to the irrational rotations of a circle, which are ergodically dense in the circle. What I've learned today is that by the definition of ergodicity, any set that behaves badly must have measure zero. If that's correct, then my NO-duplicate earth possibility is still alive. Ergodicity is a statistical attribute that describes what happens almost surely. But not absolutely surely.

This is my preliminary understanding. I no longer think ergodicity absolutely guarantees that there is a duplicate earth. If someone knows better and can walk me through the argument, I'd be grateful.

As far as your snark, it's not necessary. — fishfry

Give snark and you get snark. Fair enough?

Philosophically it's perfectly valid that you have alternative ideas about math and don't accept parts of standard modern math. But physics is most definitely based on standard modern math; and to the extent that your outlook diverges from that, you are introducing confusion into the conversation. — fishfry

Even in philosophy of maths, these are routine debates. And physics doesn't base itself on the "correctness" of maths. So you may feel confused by conversations that veer of the beaten textbook track, but there is no particular reason to think the secrets of reality are already written in those textbooks. The relationship between maths and physics is much more subtle than that.

So look, I am interested in what you have to say about standard views within mathematics. It is good that you can explain the structures of justification upon which certain mathematical positions are constructed. But if you just want to lecture on the correct methods of the academy - the stuff that will get you a pass in class - I don't really call that a conversation in the context of a philosophy forum where the thread is about the material reality of multiverses.

For the record I do understand that you have your own private notation and that you reject the standard notation of modern math, on which physics is based. Is that a fair assessment? — fishfry

Hardly. I'm quite happy to accept more than one way of doing anything. And happy also that the standard way will have proven itself to be the most pragmatic in finding the simplest path - the one that disposes of the most metaphysical baggage.

But when we are talking about how existence itself comes to be, as we are with multiverses, then that is when the axioms upon which a reductionist simplicity is founded come into metaphysical question. That is when we have to rewind and see what was being left out, or being assumed, when starting down that path.

But since we're doing physics, it's important to make sure we get the math right. — fishfry

Err, no. It is important to get the physics right and then find the right maths to make those ideas precise enough to measure.

I'm not denying that physics and maths enjoy a remarkably fruitful relation. But I do deny your understanding of how science works. The maths does not lead, it follows. And the fact that the maths then works as the precise description of the scientific claims merely shows that the maths has been suitably fitted to the task in hand. The maths was convenient.

But that's exactly wrong. You need a finite die with a very large number of faces. One for each admissible state of all the particles in some bounded region of space. — fishfry

So why did you introduce the blather about binary coin flips where you either got Earth or Mars in the course of some infinite sequence? That was the misstep I was addressing.

Of course I understood the ergodic argument that motivates the talk of repeating outcomes in a multiverse of infinite spatial extent. You were the one who had to go look it up.

And of course I understood the ridiculousness of the idea of a spherical die that could be said to "land on 1 of its infinite number of sides". It was your coin-flipping nonsense I was lampooning. Exactly which infinitesimal point are we suppose to say a sphere has landed?

Now you are talking about a finite sided die with enough degrees of freedom to represent every thermal microstate in a bounded space. As if this fixes either your blather about Earth/Mars coin-flipping sequences, or deals with my actual objections to the simplicities of the correlation-less ergodic view of nature.

Imagine your finite side die had dependencies between the faces so that all the Earth-like combos are clustered together on one neighbourhood of the die, not represented randomly across the die surface.

Perhaps now you can start to see the mathematical presumptions you are failing to take account of in attempting to construct some physical model of a "multiverse generator"? A fair die would scramble the microstates randomly. But a die more properly representing the material world, with its dependencies and entangled history, would already be built in a way that was pre-loaded. It would either fall somewhere in the neighbour where the many Earth-like outcomes were clustered - or mostly somewhere not anything like the Earth at all.

That should be blindingly obvious. Yet it's not because the idea of modelling probability spaces with correlations is not so "mainstream".

But I have zero expectation you will pick up on such a point and argue it through. You are way too intent on finding an ever higher horse from which to look down your nose at any "non-mathematician".

Why don't we agree not to interact? I was really surprised earlier that you directly replied to something I said. I don't think our interactions are productive. — fishfry

I'm amused and entertained. I'm finding the dispute productive. What more could I wish for? I'm not complaining.

As it happens I've spent the afternoon chasing down ergodicity. — fishfry

Hmm. "Ergodic" in this context is really talking about Poincare reccurence -
https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem

In an ideal gas, the particles will recross their initial conditions in the long run.

But my counter has been that the universe is more like a box of gas with its lid off. The particles are all escaping as the Universe is expanding/cooling. The locations they might want to recross may well have moved over an event horizon forever.

And then even worse, the gas is far from ideal. All the particles have correlations or dependencies. At the least, gravity and other forces are in play. So a different maths is needed to make any multiverse extrapolations.

Think more of a chaotic attractor where the trajectories could go anywhere within the bounds of the phase space, yet also cluster ... because of the internal dependencies. The system is not random but emergently self-constrained - https://en.wikipedia.org/wiki/Attractor

As it happens I've spent the afternoon chasing down ergodicity. I do know a little about it relative to the irrational rotations of a circle, which are ergodically dense in the circle. What I've learned today is that by the definition of ergodicity, any set that behaves badly must have measure zero. If that's correct, then my NO-duplicate earth possibility is still alive. Ergodicity is a statistical attribute that describes what happens almost surely. But not absolutely surely. — fishfry

Inflation generates all possible initial conditions. This means that all possible matter configurations exist in some Hubble Volume. The most likely initial conditions are those which are almost uniform with only slight variations in matter densities, which later become amplified by gravitational clustering. We inhabit a typical Hubble Volume.

The quantum mechanisms that generate the initial conditions do so effectively in the manner of an ergodic random field. What this means, is that given an ensemble of Hubble volumes, each with their own random initial conditions, then the probability distribution you get by sampling different Hubble Volumes is identical to the probability distribution you get if you sample parts of a single Hubble Volume.

It is precisely this property of ergodicity - a generic prediction of Inflation - that guarantees that everything that could have happened here, did in fact occur elsewhere, and that everything that did happen here has multiple copies.

This is my preliminary understanding. I no longer think ergodicity absolutely guarantees that there is a duplicate earth. If someone knows better and can walk me through the argument, I'd be grateful. — fishfry

You also need the Bekenstein Bound and infinity.

You are entitled to your private definition. The actual definition of an infinitesimal is a quantity x such that 0 < x < 1/n for every natural number n. — fishfry

It's actually John Wallis' definition from his Treatise on the Conic Sections, incidentally where the ∞ symbol was coined.

I think that mathematically, a coin cannot come up tails forever. There cannot not be a dup Earth given infinite space. The probability of that is 0.000... which is zero.

In an infinite sample space, Probability zero is not the same as Impossible. The term 'almost surely' was invented to cover exactly this case. It is applied to an event that is in the sample space (ie 'possible') but has zero probability.

With the usual binomial model of fair coin tossing, the event of an infinite sequence of heads is one that 'almost surely' will not occur, which is not the same as saying it cannot occur.

We can be tempted to apply this to an infinite universe and say it will Almost Surely contain duplicate Earths. To do that we need to first assume and specify a probability distribution for the configuration of mass-energy across the universe. I imagine that can be done, but it's too long to get into in this post.

In an infinite sample space, Probability zero is not the same as Impossible. The term 'almost surely' was invented to cover exactly this case. It is applied to an event that is in the sample space (ie 'possible') but has zero probability. — andrewk

I realized that I had made an error and backed off my 'certainly' claim pending a redo. In the end, I decided that no-Earth was not in sample-space. It's not the same as stabbing at an infinite list of impossible to hit things, inevitably hitting one of them.
That math might not be right either (I didn't show it), but it seems to hold the probability of hitting a specific countable number in random sample of the uncountable reals.
I am basing a lot of my claims on another thread debating why 0.999... is 1, not just infinitesimally close to it. It was explained by someone who knows their stuff far better than I.

With the usual binomial model of fair coin tossing, the event of an infinite sequence of heads is one that 'almost surely' will not occur, which is not the same as saying it cannot occur. — andrewk

Not sure if a model of fair coin tossing applies. In an uncountable sample space, the one you actually hit cannot be represented by any number of coin tosses.

The set of possible outcomes from an infinite sequence of coin tosses is uncountable. It has a natural one-to-one correspondence with the real numbers in the interval [0,1], via their binary representation, where a Head (Tail) on the n-th toss is interpreted as a 1 (0) in the n-th position after the dot (comma for Europeans). An infinite sequence of pure Heads (Tails) maps to 1 (0) - the upper (lower) bound of the interval.

I am basing a lot of my claims on another thread debating why 0.999... is 1, not just infinitesimally close to it. It was explained by someone who knows their stuff far better than I. — noAxioms

Hah. There certainly is an official position on this. But it is more about what has to be agreed to make the maths come out right than one based on force of metaphysical argument.

And I'm not complaining. Maths needs to secure its constructs. It needs to be axiomatic.

I'm just reminding that this is what happens and so maths isn't in a position to tell metaphysics "what is really going on" due to what if finds works. Maths can act as a powerful constraint on free metaphysical speculation, and also serve as a powerful inspiration to further inquiry. But it isn't how metaphysical truth is discovered.

So the infinite and the infinitesimal speak to the need to establish limits. Constructive actions like counting need to have constraints to bound them as well. It is in fact the same dialectical issue which is at the root of metaphysical reasoning. For something to be the change, something complementary must be made the bit that stands still.

The number line has to be both continuous yet discrete at the same time. It must be a line composed of points. So of course some fancy mathematical machinery must be added to negotiate what must be a tricky change-over going on somewhere. What connects the points? What permits an exact cut?

The official answer works. But it is also pervaded by a spirit of "OK guys, shut your eyes for a moment, don't ask any annoying questions, as we do this bit of tricky surgery".

The set of possible outcomes from an infinite sequence of coin tosses is uncountable. — andrewk

Right. Can count only the finite ones (trivially at that). The coin model works, and thus 1 followed by all zeros is possible. Shot down again.

Am still enjoying the concept. Perhaps a proof that no copy is possible then? The distant Earth might be outside our causal cone now, but it wasn't always. All matter (or whatever it was back then) was at the big bang and is part of the causal history of this planet. Since information is preserved, a perfect copy has it all. Thin ice there. Maybe the info was shipped out of reach, even if preserved.
Anyway, if that works, Earth (or just an apple say), contains the history of its entire causal cone in the past and cannot be recreated without that entire past, which is infinite stuff.
Tegmark assumes Earth can be represented with finite state, and computes the distance needed to get the probability of a repro up to about 1. If if the state is no finite number, then no copy.

The number line has to be both continuous yet discrete at the same time. — apokrisis

I just picked this out. Agree with your post. My history-of-everything assumes no discreetness at all. Any tiny difference way below Planck constant would still yield a measurable difference after chaos gets to do its thing. Sort of invalidates the Planck concepts.
I hate discreet anything. How slow can something go? If it moves one Planck length each hour, what does it do while it waits for the next one? A discreet universe would exhibit jaggies. Ewww....

Let's bring down the scale to a manageable size to explore this concept. Instead of Earth, let's consider one hydrogen atom. Now, an atom has a finite number of configurations, or states that it can possibly be in (10 for hydrogen, I think). Since there are obviously a lot more hydrogen atoms than that, there have to be a lot of atoms that are absolutely identical, regardless of their prior histories.

You can see how this can be scaled up by adding more atoms and particles to the system: they each have some finite number of states, and so do their combinations, even allowing for interactions. The number of degrees of freedom rises dramatically as you expand outwards, but the principle remains the same.

I won't vouch that the math actually works out for macroscopic systems, i.e. that the number of possible states increases slower than the size of the system, but that is the argument that Vilenkin and some others make.

Am still enjoying the concept. Perhaps a proof that no copy is possible then? The distant Earth might be outside our causal cone now, but it wasn't always. — noAxioms

Our visible universe may well have been the size of a point at the Big-Bang, but the entire Level 1 Multiverse was not. The Multiverse was as infinite then as it is now.

Inflation guarantees the immediate creation of causally disconnected regions.

Tegmark assumes Earth can be represented with finite state, and computes the distance needed to get the probability of a repro up to about 1. If if the state is no finite number, then no copy. — noAxioms

As I've mentioned several times, the Bekenstein bound severely limits the number of states available to any volume of space.

So we have:

Infinite universe+Inflation+all initial conditions+ergodicity+Bekenstein Bound = guaranteed copies of Hubble Volumes.

Thanks guys. Didn't think it would work. A few holes to point out, but in the end, the state of Earth does not require all the universe to have been this and thus.

Now, an atom has a finite number of configurations, or states that it can possibly be in (10 for hydrogen, I think). — SophistiCat

One atom has no position, velocity, or other relations. But a group does, and each atom has innumerable additional states that make up its relationships with the others. Really innumerable??? Maybe not.

Our visible universe may well have been the size of a point at the Big-Bang, but the entire Level 1 Multiverse was not. — tom

Maybe my model is incorrect, but this seems wrong. Since the level-1 spheres overlap, they're all points in the beginning, and all the same point at that, else they'd not overlap. I don't totally grasp eternal inflation theory, where perhaps the inflation stuff rips away as normal space forms in the bubble, but that is not a description of a point except the point where the bubble first began, not necessarily being the point that represents our hubble sphere.

As I've mentioned several times, the Bekenstein bound severely limits the number of states available to any volume of space. — tom

Limits it given finite energy. If the initial infinite universe was actually a point, there is infinite energy/information there. But this actually kills my idea. Earth is a limited space with limited energy. The bound applies. Earth cannot be in a unique state that requires the history of the entire set of material that was once in its causal past. Tegmark was working on a bound such as this, and then just computed how much space was required to make it likely that a good majority of those (valid) states were realized.

Information must be preserved, but like other conserved things, it need not all be conserved here. Most of that information is shipped off elsewhere.

Maybe my model is incorrect, but this seems wrong. Since the level-1 spheres overlap, they're all points in the beginning, and all the same point at that, else they'd not overlap. — noAxioms

How much expansion is required to produce a literally infinite universe from a point in a mere 13.8 billion years?

Why do you think Hubble Volumes were ever in contact or overlap?

Limits it given finite energy. If the initial infinite universe was actually a point, there is infinite energy/information there. But this actually kills my idea. Earth is a limited space with limited energy. The bound applies. Earth cannot be in a unique state that requires the history of the entire set of material that was once in its causal past. — noAxioms

The initial infinite universe cannot have been finite. No physical process can take something finite and make it infinite in finite time.

The initial energy density of the universe was also finite.

There are also Level 2 multiverse earths.

How much expansion is required to produce a literally infinite universe from a point in a mere 13.8 billion years? — tom

The comment here only makes sense if interpreted as sarcasm. It implies that there might have been finite hubble volumes, and after enough time, that goes to infinite. The greater the expansion, the less time it takes to do this. No, not my view.
The way I see it: If the geometry is such that the universe wraps (like the sphere of the balloon analogy), then there are finite Hubble-volumes. Assuming not, then if the expansion rate is increasing, there are infinite such volumes. If the rate is not increasing, light will eventually get from anywhere to anywhere else, and the universe is a single Hubble volume. At no point does "13.8 billions years" play into that.

Why do you think Hubble Volumes were ever in contact or overlap?

Why do you think they don't? We are at the exact center of our Hubble volume. Isn't that amazing? From the perspective of a planet 10 BLY away to the left (all this is in comoving coordinates BTW), they are centered on a different volume that encompasses us way to the right. Their volume ends further to the right of us, but not a whole lot further. Some distant galaxy to our right can be seen from here but can never ever be seen by them. It is outside their Hubble Volume. Our volumes overlap else we couldn't see each other.
To say they're all nonoverlapping implies there are discreet chunks of disjoint space with one preferred point in each of them which is their center. My model doesn't look like that.

There are also Level 2 multiverse earths.

I would think so, yes. Level 4 as well.

Why do you think Hubble Volumes were ever in contact or overlap? — tom

Another note: Level 3 universes overlap as well. There is amazing symmetry between the level 1 and level 3 concepts.
Level 1 is just Schrodinger's box implemented with distance instead of technology.

Do you have any references on what the measure preserving transformation is? I mean, if we're speaking about ergodicity, it has to be the ergodicity of a measure preserving transformation. Another way of putting it is what is a 'step' in the 'trajectory of the universe' defined as? And how can it be established as ergodic?

Another thing - how can ergodicity be used to show not just that the long term probability of set visitation is nonzero, but that its arrival time is finite? There's a distinction in terms of finite Markov chains, having an infinite arrival or 'revisitation' time excludes a state from being ergodic (and thus the chain from being ergodic in terms of all states).

You can see how this can be scaled up by adding more atoms and particles to the system: they each have some finite number of states, and so do their combinations, even allowing for interactions. The number of degrees of freedom rises dramatically as you expand outwards, but the principle remains the same. — SophistiCat

Do those combinations also include how they are arranged together in space? If space were continuous, then there would seem to be an infinite number of possible ways I can put them together. I can imagine two atoms being separated by 1 nm, 2 nm or any 1/n nm, for example. I am not sure if it's physically possible but I can imagine it in my head. Or does that not matter?

The comment here only makes sense if interpreted as sarcasm. It implies that there might have been finite hubble volumes, and after enough time, that goes to infinite. The greater the expansion, the less time it takes to do this. No, not my view. — noAxioms

So, what rate of expansion do you think might be required to turn a subatomic spec into a literally infinite universe in 13.8 billion years? Have you done the calculation?

The way I see it: If the geometry is such that the universe wraps (like the sphere of the balloon analogy), then there are finite Hubble-volumes. Assuming not, then if the expansion rate is increasing, there are infinite such volumes. If the rate is not increasing, light will eventually get from anywhere to anywhere else, and the universe is a single Hubble volume. At no point does "13.8 billions years" play into that. — noAxioms

Hubble Volumes are finite. If the "universe wraps" i.e. it is a finite 3-sphere, I think all bets are off for Level 1 (and Level 2) multiverse statistics.

The rate of expansion may be static, increasing, or decreasing. As long as there is a +ve Hubble constant, there will be Hubble Volumes.

Why do you think they don't? We are at the exact center of our Hubble volume. Isn't that amazing? From the perspective of a planet 10 BLY away to the left (all this is in comoving coordinates BTW), they are centered on a different volume that encompasses us way to the right. Their volume ends further to the right of us, but not a whole lot further. Some distant galaxy to our right can be seen from here but can never ever be seen by them. It is outside their Hubble Volume. Our volumes overlap else we couldn't see each other.
To say they're all nonoverlapping implies there are discreet chunks of disjoint space with one preferred point in each of them which is their center. My model doesn't look like that. — noAxioms

Sure, your Hubble volume and my Hubble volume might be slightly different in 14 billion years. In the mean time, there are an infinite number of Hubble volumes that were never in causal contact with ours.

So, what rate of expansion do you think might be required to turn a subatomic spec into a literally infinite universe in 13.8 billion years? Have you done the calculation? — tom

You persist with this. Is it a serious question? 6 days, after which enough expansion took place to qualify as infinite. On the 7th day, the expansion rested. I really don't know how else to answer that.

The rate of expansion may be static, increasing, or decreasing. As long as there is a +ve Hubble constant, there will be Hubble Volumes.

A Hubble volume is not a type-1 universe. It is just the volume containing the matter whose distance from us is growing at sub-lightspeed. The Type-1 universe is bounded by the event horizon, beyond which things cannot ever have a causal effect here. It is something like 16BLY in radius at this time (comoving coordinates again).
About the Hubble constant, it is not a constant since the expansion rate is increasing. The constant is just what the value is now, and thus can only be measured to so much precision.
I looked at its description on space.com and found text more worthy of the New York Post:

As of March 2013, NASA estimates the rate of expansion is about 70.4 kilometers per second per megaparsec. A megaparsec is a million parsecs, or about 3.3 million light-years, so this is almost unimaginably fast. — space.com

km/sec per megaparsec is not a velocity, so not sure how this could be unimaginably fast. 70km/sec is not much more than the orbital speed of Mercury, and I think I can manage the imagination of it. Sorry. I was hoping for better from a site like that.

Sure, your Hubble volume and my Hubble volume might be slightly different in 14 billion years. In the mean time, there are an infinite number of Hubble volumes that were never in causal contact with ours. — tom

The example was about the nearby overlapping ones, not the countless more distant ones.

You persist with this. Is it a serious question? 6 days, after which enough expansion took place to qualify as infinite. On the 7th day, the expansion rested. I really don't know how else to answer that. — noAxioms

Maybe you should show your working? Given an initial 1m3 of space-time, what expansion rate is required to turn it into literally infinite volume in any finite time?

Since the level-1 spheres overlap, they're all points in the beginning, and all the same point at that, else they'd not overlap.

I haven't gone through this idea carefully, but I'm moderately confident there is no 'reasonable' mathematical model in which a spatially infinite universe contains a time zero. If that's correct then there is no question of whether the universe was infinite or a single point at that time, since there is no such time.

We can create a model in which the spacetime contains all times after time zero but does not contain time zero itself. The earlier the time (The smaller its time coordinate), the greater the universe density becomes, so that it increases without limit as t approaches zero. If the universe is spatially infinite now, it will be spatially infinite at all those times too, no matter how small t is.

When I say 'no reasonable model' I mean that a model that included time zero would have to be discontinuous in most significant respects at that time, in which case there's really no point in including time zero in the model since it would have no causal connection with the rest of the spacetime.

These problems do not arise with a spatially finite universe.

In practice, we don't need to worry about a time zero for either a spatially finite or a spatially infinite universe, because the General Theory of Relativity, which is used to do the backwards projection, loses validity as the scale becomes very small, and we have no theory to replace it. We can't use quantum mechanics because it ignores gravitational effects and in a very dense universe those cannot be ignored.

Maybe you should show your working? — tom

I did. I took the question for sarcasm and responded in kind when you persisted.

Given an initial 1m3 of space-time, what expansion rate is required to turn it into literally infinite volume in any finite time?

Not going to happen. Universe was never 1m3 it seems.

I haven't gone through this idea carefully, but I'm moderately confident there is no 'reasonable' mathematical model in which a spatially infinite universe contains a time zero. If that's correct then there is no question of whether the universe was infinite or a single point at that time, since there is no such time. — andrewk

If the model has a event 0, there is no space to have a size. That's what makes it a singularity.

In practice, we don't need to worry about a time zero for either a spatially finite or a spatially infinite universe, because the General Theory of Relativity, which is used to do the backwards projection, loses validity as the scale becomes very small, and we have no theory to replace it. We can't use quantum mechanics because it ignores gravitational effects and in a very dense universe those cannot be ignored. — andrewk

I don't know my cosmology enough to describe the actual workings of our big bang. Inflation theory says there was different physics for a short time, low temperature, and perhaps the usual notions of 'density' wouldn't apply. The mass of the universe, if existing in some sort of finite volume, would form a black hole and never get off the ground.

Suffice it to say that for the purposes of this thread, the declaration that space is infinite implies it was always infinite ever since it was space. The material/energy probably never fully interacted. There is stuff that never was part of our causal history, even going all the way back. No reason to worry about it anyway since the Earth contains finite state and thus has no need of that history to be identical to another Earth.
We do have history though. If an exact duplicate Boltzmann Earth suddenly popped into existence in orbit around a plausible star, I would be on it and that 'me' would notice damn fast that it wasn't the native Earth.

We can create a model in which the spacetime has all times after time zero but does not contain time zero itself. The earlier the time (The smaller its time coordinate), the greater the universe density becomes, so that it increases without limit as t approaches zero. — andrewk

Not according to Inflation, the theory on which the Level 1 and Level 2 Multiverses are premised.

Suffice it to say that for the purposes of this thread, the declaration that space is infinite implies it was always infinite ever since it was space. The material/energy probably never fully interacted. — noAxioms

Precisely! No finite physical process can create something infinite out of something finite in finite time whether you choose to show your workings or not.

Edit: Just to point out that to travel between Level 2 multiverses, you would need to traverse the part of the infinite space-time that our universe emerged from. This would be extremely difficult as these regions of the Master Space-Time are still undergoing inflation. As if travelling between Level 1 multiverses were not difficult enough.

I did. I took the question for sarcasm and responded in kind when you persisted.

Given an initial 1m3 of space-time, what expansion rate is required to turn it into literally infinite volume in any finite time?

Not going to happen. — noAxioms

You are passing up a valuable learning opportunity! Go on, give it a try!

Not going to happen.
— noAxioms

You are passing up a valuable learning opportunity! Go on, give it a try! — tom

I mean the 1m3 expanding to infinity is not going to happen. OK, I worded it ambiguously, and you took it to mean that I'm not going to attempt the math.

Inflation generates all possible initial conditions. — tom

That's very interesting. How does it know to do that? In the early moments of the universe it's cranking out all these possible configurations, and it's only got one more left. How does it know that? What if it forgets to do one particular configuration? Can it go back and do it later? Can the universe continue to exist or does this one single imbalance make the universe unstable in some way?

This is my preliminary understanding. I no longer think ergodicity absolutely guarantees that there is a duplicate earth. If someone knows better and can walk me through the argument, I'd be grateful.
— fishfry

You also need the Bekenstein Bound and infinity. — tom

So my understanding is correct? I'm gratified. "Almost surely" is not the same as "surely." All these statistical arguments leave the possibility of exceptions, unlikely as they may be.

Now the Beckenstein bound I believe has something to do with relating the energy in a bounded region of space to its information carrying capacity. It takes a certain amount of energy to change the state of a system so that the collection of all possible state changes is finite and can be calculated. Is that about right?

So how do we get from there to duplicate earths?

You know I just don't believe this duplicate earth story. Say there's a universe or a multiverse and it's got every possible state represented infinitely many times ... except there is one state that just happens to only occur once, by incredible amazing luck ... one little blue watery planet with bad politics, third from the sun ... and it's the only one like it in the entire multiverse.

I just don't see why such a universe couldn't exist. It would still have all the required statistical properties. But if your universe is infinite, then "almost surely" is NOT the same as "surely" and individual exceptions to the statistical properties may exist. That's one of the limitations of extending probability theory to infinite spaces. You lose certainty.

But we're making progress if you agree that ergodicity by itself is not sufficient.