How many infinite universes can fit within a multiverse?

[Athanasius]

In multiverse cosmological models, what sort of infinities are usually used? For instance, in many models is it thought that the multiverse is a continuum - an unlistable (uncountable) infinite set comparable to the real numbers, and that it contains universes of Aleph_0 listable infinities, comparable to the set of natural numbers? And if so, how would we derive an infinite number of listable infinities from a single unlistable one, if the universes all arise from quantum fluctuations within the same field? It seems that a dimensional quantum or vacuum field can contain only so many infinities within it's dimensions. Would not just one take up all of the room in the field dimensions it occupies? It seems that if even if two are an infinite distance apart from each other, they should overlap because they are infinite.

 

[PeterDonis]

in many models is it thought that the multiverse is a continuum

In *all* models that have any experimental confirmation, the multiverse is a continuum. And so is each universe within it.

and that it contains universes of Aleph_0 listable infinities, comparable to the set of natural numbers?

No. Our universe, for example, is modeled as a continuum in all current models that have been tested experimentally. The only attempt I'm aware of to model the universe as anything other than a continuum is loop quantum gravity, but that's speculative.

It seems that a dimensional quantum or vacuum field can contain only so many infinities within it's dimensions.

Don't confuse dimensionality with infinite cardinality. A one-dimensional line segment one picometer long, from the standpoint of our mathematical models, has the same infinite cardinality (the continuum) as the entire 10- or 11-dimensional spacetime that is used in string theory.

It seems that if even if two are an infinite distance apart from each other, they should overlap because they are infinite.

Don't confuse infinite cardinality with infinite distance. "Distance" is defined in terms of a metric, not in terms of "number of points". It's perfectly possible for non-overlapping regions of a finite size, in terms of the metric, to have an infinite continuum of points in them. See above.

 

[Athanasius]

Hi PeterDonis, thanks for that information. It makes sense to me that the universes within a multiverse continuum would have to be continuums themselves, if you were to have discrete, non-overlapping universes that occupy the same field dimensions. Using a one-dimensional continuum line that extends to infinity in both directions, if I were to discern a countable infinity inside it, than I have to use some metric to count, such as {-1, 0, 1}. From all of the elements of that set, I could create more infinite sets, such as {...-2, 0, 2...}. Like the parent set, this second countable child infinity would extend across the entire continuum. I could create more discrete countable child infinities within it, such as {...-1, 1, 3..}, but I would have to jump over elements of the first set to count that set. Those two discrete sets alone would contain all of the elements in the parent set. So there is a limit to the number of discrete sets (sets that do not share the same elements) that can exist within a parent set. In fact, all other sets that contain odds and evens (which would be all possible child sets) will contain elements of these first two child sets. So it does not seem to me that you could have an infinite number of discrete countable sets within a continuum. I have an appointment now, but I will share a little more of my thoughts concerning some additional implications of this later today. Meanwhile, If you and others could critique my logic here, I would appreciate it.

 

[PeterDonis]

if you were to have discrete, non-overlapping universes that occupy the same field dimensions.

I don't understand what this means. What are "field dimensions"?

Using a one-dimensional continuum line that extends to infinity in both directions, if I were to discern a countable infinity inside it

What would that mean, physically? Of course I can find a countably infinite subset of any continuum, but so what? What physics does that correspond to? I think you are throwing around a lot of mathematics here that doesn't have any physical meaning. You're also getting the math wrong; see below.

So there is a limit to the number of discrete sets (sets that do not share the same elements) that can exist within a parent set.

This is true if the "parent set" is countable, yes; you can only partition a countable infinity into a finite number of countably infinite subsets. (The finite number can be arbitrarily large, however; instead of partitioning by odd/even, which corresponds to taking equivalence classes modulo two, you could take equivalence classes modulo a million, which would give a million disjoint countably infinite subsets, or modulo a trillion, or a googol, or a googolplex, or any finite number you like, as large as you like.)

However, your logic here is wrong if the "parent set" is a continuum.

it does not seem to me that you could have an infinite number of discrete countable sets within a continuum.

Yes, you can, because a continuum is not countable, and the logic you used (which basically amounts to using properties of modulo arithmetic, as I noted above) does not work if the parent set is not countable. You can have an infinite number of disjoint, countably infinite subsets of a continuum. In fact, you can have an *uncountably infinite* number of disjoint, countably infinite subsets of a continuum.

 

[Athanasius]

I don't understand what this means. What are "field dimensions"?

Forgive me for using unconventional terminology. Doesn't any scalar or quantum field by definition have defining dimensions, and don't fluctuations occur within certain of those dimensions? That is what I meant by field dimensions.

What would that mean, physically? Of course I can find a countably infinite subset of any continuum, but so what? What physics does that correspond to? I think you are throwing around a lot of mathematics here that doesn't have any physical meaning. You're also getting the math wrong; see below."?

In this case, I was referring to whatever metric is chosen to uniformly designate the points in the quantum field. By doing that, wouldn't we end up with a countable subset of points in the multiverse continuum? However, I think I see your point here. Though perhaps using that metric we could "count" the volume of the universes, we could not count the number of them, if they vary in size, as we would expect in a continuum. But that brings up another question.

If these universes have any order, definition or regularity to them, they must have countable elements such as atoms or elementary particles within them. These must be measurable by some countable metric. Could not that countable metric be extended across the entire continuum? If each of these universes is infinite according to that metric, whatever it may be, how could they vary in size? If they occupy the same dimensions within the quantum field as other universes, how could they be disjoint or discrete? So it is hard for me to see to see how you could have an infinite number of infinite universes with any sort of order to them within a continuum, unless you have infinite dimensions. If I have missed something in my reasoning here, please let me know.

 

[PeterDonis]

Doesn't any scalar or quantum field by definition have defining dimensions

Not really. Spacetime has dimensions, and the abstract spaces in which gauge fields are defined have dimensions; but the fields themselves do not. I think you may be confusing "dimensions" with the transformation properties of the field: fields can be scalars, vectors, tensors, spinors, etc., but these refer to how the field changes under various transformations, like rotations and boosts. They have nothing to do with the dimensionality of the underlying manifold.

In this case, I was referring to whatever metric is chosen to uniformly designate the points in the quantum field.

That's not what a metric does. You can't use a metric to designate points: you have to have points first, before you can even define a metric. A metric is a function that, given a pair of points and a curve connecting them, tells you the length of the curve between the two points. The curve is a continuum.

By doing that, wouldn't we end up with a countable subset of points in the multiverse continuum?

No.

Though perhaps using that metric we could "count" the volume of the universes

If the universe even has a finite volume. Many models of universes have infinite volumes. And volumes are real numbers, i.e., they are continuous, so "count" is not really a proper word to describe how you measure them.

we could not count the number of them, if they vary in size, as we would expect in a continuum.

The number of universes has nothing to do with their size. You could have multiple universes all of the same size. You could even have a continuum of different universes all of the same size; they would just continuously vary in other parameters.

If these universes have any order, definition or regularity to them, they must have countable elements such as atoms or elementary particles within them.

Why must there be "elements" at all? I know all of our current models assume there are, but why *must* there be? And why must the number of such elements be countable? Don't confuse "countable" with "discrete" or "quantized". Atoms and elementary particles have quantized properties like mass and spin; the values of these properties are discrete and don't form a continuum. But that doesn't mean there must be a countable number of atoms or elementary particles in the universe. Many models assume that there is, but that's an extra assumption; it isn't made necessary just by the fact that atoms and elementary particles have discrete values for some of their properties.

These must be measurable by some countable metric.

How does this follow? Even if atoms or elementary particles have discrete values for some properties, that does not mean they must have discrete values for all properties. Some properties are continuous even when applied to atoms or elementary particles: for example, position, momentum, and energy.

Could not that countable metric be extended across the entire continuum?

How can a countable metric possibly "measure" an uncountable infinity of possible values?

If each of these universes is infinite according to that metric, whatever it may be, how could they vary in size?

I'll leave out "according to that metric", since your assumption about what kind of metric it is is incorrect (see above). But your question is still valid if we leave that phrase out. The answer is, they wouldn't. They would vary in other parameters.

If they occupy the same dimensions within the quantum field as other universes

The quantum field itself doesn't have dimensions. See above. Obviously whatever "space" or "multiverse" or whatever you want to call it, whatever it is that contains all the universes, must have enough "dimensions" to contain them all, to the extent that even has meaning. All the different universes could just exist without having any "dimension" that connects them.

it is hard for me to see to see how you could have an infinite number of infinite universes with any sort of order to them within a continuum, unless you have infinite dimensions.

This doesn't follow either. To see why, consider a corresponding claim: it is impossible to have an infinite number of infinite planes within a continuum, unless you have infinite dimensions. This is obviously false: a continuous infinity of continuously infinite planes can be contained in ordinary three-dimensional Euclidean space. Similarly, a continuous infinity of continuously infinite three-dimensional spaces could be contained in a four-dimensional space; you wouldn't need infinite dimensions. Similar reasoning applies even if each universe has ten or eleven dimensions, as string theory currently says; you could have a continuous infinity of them in an eleven- or twelve-dimensional space. You wouldn't need infinite dimensions.

 

[Athanasius]

Not really. Spacetime has dimensions, and the abstract spaces in which gauge fields are defined have dimensions; but the fields themselves do not. I think you may be confusing "dimensions" with the transformation properties of the field: fields can be scalars, vectors, tensors, spinors, etc., but these refer to how the field changes under various transformations, like rotations and boosts. They have nothing to do with the dimensionality of the underlying manifold.

I am not sure I understand you here. Are you simply pointing out the difference between abstract concept and reality, or are you saying that a field can exist without any property that we could abstractly conceive of as dimensionality?

That's not what a metric does. You can't use a metric to designate points: you have to have points first, before you can even define a metric. A metric is a function that, given a pair of points and a curve connecting them, tells you the length of the curve between the two points. The curve is a continuum.

If the universe even has a finite volume. Many models of universes have infinite volumes. And volumes are real numbers, i.e., they are continuous, so "count" is not really a proper word to describe how you measure them.

I realize that. Perhaps designate was not a clear word to use. I did not mean "designate" in terms of creating, but rather in terms of selecting already existing points or elements, each a uniform distance apart. Furthermore, when I used the word "countable" I did not mean to say that you could actually count an infinite set. What I meant was "countable" or "listable" in the sense that it is used with infinite sets, i.e, that you could list the elements of the set up in one to one correspondence with the set of natural numbers. In other words, an aleph_0 infinity. So the idea I was expressing was that if you define an infinite metric space across that continuum, you are selecting points that are a uniform distance apart from each other, and the set of those points only is an aleph_0 infinity that is contained within the continuum.

The number of universes has nothing to do with their size. You could have multiple universes all of the same size. You could even have a continuum of different universes all of the same size; they would just continuously vary in other parameters.

Ok, I understand.

Why must there be "elements" at all? I know all of our current models assume there are, but why *must* there be? And why must the number of such elements be countable? Don't confuse "countable" with "discrete" or "quantized". Atoms and elementary particles have quantized properties like mass and spin; the values of these properties are discrete and don't form a continuum. But that doesn't mean there must be a countable number of atoms or elementary particles in the universe. Many models assume that there is, but that's an extra assumption; it isn't made necessary just by the fact that atoms and elementary particles have discrete values for some of their properties.

How does this follow? Even if atoms or elementary particles have discrete values for some properties, that does not mean they must have discrete values for all properties. Some properties are continuous even when applied to atoms or elementary particles: for example, position, momentum, and energy.

Even if only some of the properties are countable, that is enough to make my point. And even if not all universes contain countable elements (perhaps chaotic, high entropy ones might not), if you have a certain percentage that do, there will be an infinite number of universes containing countable elements.

How can a countable metric possibly "measure" an uncountable infinity of possible values?

Of course, there is no way to measure an infinity. That is not what I meant. I was just referring to an aleph_o subset of the units measured by the metric adopted.

I'll leave out "according to that metric", since your assumption about what kind of metric it is is incorrect (see above). But your question is still valid if we leave that phrase out. The answer is, they wouldn't. They would vary in other parameters.

OK, that makes sense to me.

This doesn't follow either. To see why, consider a corresponding claim: it is impossible to have an infinite number of infinite planes within a continuum, unless you have infinite dimensions. This is obviously false: a continuous infinity of continuously infinite planes can be contained in ordinary three-dimensional Euclidean space. Similarly, a continuous infinity of continuously infinite three-dimensional spaces could be contained in a four-dimensional space; you wouldn't need infinite dimensions. Similar reasoning applies even if each universe has ten or eleven dimensions, as string theory currently says; you could have a continuous infinity of them in an eleven- or twelve-dimensional space. You wouldn't need infinite dimensions.

Thanks for correcting me; that makes sense to me as well. These would be what are called branes, correct? I had read about the concept before, I just did not think of it before making my post, and apologize for making it too hastily.

Thanks for your replies, Peter! You have made this much clearer for me. What would you say the odds are that a universe within a given brane will be infinite is?

 

[PeterDonis]

Are you simply pointing out the difference between abstract concept and reality, or are you saying that a field can exist without any property that we could abstractly conceive of as dimensionality?

I'm saying that fields, as they are modeled in physics, don't have dimensionality; dimensionality is possessed by other mathematical entities in the models, but not by the fields.

I did not mean "designate" in terms of creating, but rather in terms of selecting already existing points or elements, each a uniform distance apart.

That's fine, but not all points are a uniform distance apart, so I'm not sure what you are accomplishing by picking out a particular set of points that are each a uniform distance apart. Such a set will be countable, yes, but so what? There are an uncountable infinity of points that are not in the set.

when I used the word "countable" I did not mean to say that you could actually count an infinite set.

Of course not. "Countable" just means an infinite set with cardinality $\aleph_0$. That's understood.

if you define an infinite metric space across that continuum, you are selecting points that are a uniform distance apart from each other

No, you're not. The act of picking out this particular set of points is not part of defining the metric, and is not logically required by defining the metric. The metric makes it possible to pick out this particular set of points, but the act of picking them out is still a separate thing from defining the metric.

Even if only some of the properties are countable, that is enough to make my point.

Then I don't understand what your point is, because I don't see how some of the properties being countable proves anything.

even if not all universes contain countable elements (perhaps chaotic, high entropy ones might not), if you have a certain percentage that do, there will be an infinite number of universes containing countable elements.

Possibly, possibly not. There could be only a finite number of universes containing countable elements, even if the total number of universes is uncountably infinite.

These would be what are called branes, correct?

Not really. In string theory, branes are higher-dimensional analogues of strings--strings are one-dimensional, branes are anything from two- to nine- or ten-dimensional (depending on whether spacetime as a whole has ten or eleven dimensions). They are all contained within a single spacetime, i.e., within a single universe (although this universe has ten or eleven dimensions, not just the four we observe). The extra dimension that would enable a "multiverse" to contain a continuous infinity of universes is in addition to all that.

Some discussions of branes view what we observe as our "universe", i.e., the 4-dimensional spacetime that we observe, as a brane within the ten- or eleven-dimensional spacetime of string theory. This then leads to speculations about there being other 4-dimensional spacetimes, other branes, within the ten- or eleven-dimensional spacetime of string theory, with these other 4-branes being separated from the one we observe in other dimensions. I'm not sure what the current status of such speculations is in string theory, but AFAIK the other 4-branes in this model are not really other "universes"; they're just aspects of our universe that we can't directly observe, but can indirectly contribute to things we can observe, like the strength of gravity.

Thanks for your replies, Peter! You have made this much clearer for me.

You're welcome! I'm glad our discussion has helped.

What would you say the odds are that a universe within a given brane will be infinite is?

I'm not even sure this question is well-defined, since I don't know of any way to put a probability measure on the set of all possible universes.