# Do infinities exist in nature?

### by Marianne Freiberger and Rachel Thomas

What would you see if you came to the edge of the Universe? It's hard to imagine so it's tempting to conclude that the Universe doesn't have an edge and therefore that it must be infinite. That's not a necessary conclusion however. There are things that are finite in extent but still don't have an edge, the prime example being the surface of a sphere. It's got a finite area but when you walk around on it you'll never fall over an edge. The question of whether the Universe is finite or infinite is one that still hasn't been answered, and there are mathematical models that allow for both possibilities. More generally, the question of whether any infinite quantities can arise in the Universe is a deep one. In April this year philosophers, cosmologists and physicists came together at the University of Cambridge, as part of a conference series on the philosophy of cosmology, in order to discuss it. *Plus* went along to find out more (and you can also listen to the interviews we did in our podcast).

### Infinity that doesn't bite

John D. Barrow

People have been studying infinity and its relation to reality for a long time. "The idea of studying infinities in physics really began with Aristotle," says the Cambridge cosmologist John D. Barrow. "Aristotle made a clear distinction between two types of infinity. One he called *potential infinities* and he was quite happy to allow for those to appear in descriptions of the world. These are just like lists that never end. The ordinary numbers are an example; one, two, three, four, five, and so on, the list goes on forever. It's infinite, but you never reach or experience the infinity. In a subject like cosmology, there are lots of infinities like that and most people are quite happy with them.
For example, the Universe might have infinite size; it might have an infinite past age, it might be destined to have an infinite future age. These are all potential infinities, so they don't bite you as it were, they're just ways of saying that things are limitless, they're unbounded, like that list of numbers."

While most people are happy to accept that potential infinities may exist, we still don't know whether they actually do. "When you look at the Universe, how far you can see is strictly limited, because the Universe has been in existence for a finite time, for around 14 billion years," says George Ellis, a cosmologist from the University of Cape Town. "Light travels at the speed of light, so you can only see out to a distance essentially of 14 billion light years; it's a little bit bigger but basically that's it. There's no way you can see to infinity. It's like [looking out] from a tower on the surface of the Earth; you can see to the horizon, but you can't see beyond. In that case you can get on a plane and fly to the other side. In the case of the Universe, the scale is such that we can't move; we’re stuck at one point and we can only see the Universe from one point out to a finite distance."

But even the finite past Ellis alludes to, the 14 billion years the Universe has been in existence, is more a figure of speech than a certainty. We know that the Universe is currently expanding and if we follow that expansion backwards we come to a special point in time, the Big Bang, which we take to be the beginning of our Universe. But the accepted theories of physics, Einstein's general theory of relativity and quantum physics, don't apply at that moment. We don't have a definite theory that can describe it, only a host of candidate theories. "Some such theories say there was no beginning and others say there was," says Ellis. "Basically we are making educated guesses. We can't do the experiments that will tell us which is correct because we can't get to the energies that are big enough."

George Ellis

While the moment of the Big Bang is beyond the reach of current theories, there is a widely accepted model that kicks in just moments after. It's called *inflation*. Anthony Aguirre, from the University of California at Santa Cruz, believes that it can tell us something about the extent of the Universe. "Inflation is the idea that at an early time the Universe expanded exponentially, so it doubled in size 100 times or something like that, at least during some very short period of time. What people understood early on was that the inflationary theory gives a whole bunch of suggestive predictions, many of which have come true and many of which will be tested in upcoming experiments. That gives us a lot of confidence in inflation, but it also has very interesting side effects."

One of these side effects is that inflation might have gone on at different rates in different regions of the Universe. In some region, the rapid doubling in size will have stopped after a while, resulting in a region of observable Universe like ours. In other regions though, because of spatial variations in the make up of the universe, inflation might go on forever. "You have an infinite spacetime not because you've postulated spacetime is infinite, but because you thought of a process that naturally leads to an infinite spacetime," says Aguirre. "I think that's a very interesting difference, because you can test that process in other ways." If your tests make you believe that this is what actually happened, then the infinity of spacetime pops out as a result of a consistent theory.

Intriguingly, theory also suggests that the extent of space and time depend on your view point. With his general theory of relativity Einstein told us that time and space are inextricably linked, hence the term *spacetime*. If you want to say something about space or time separately, you need to chop that spacetime up mathematically. "It turns out that even questions like 'Is space finite or infinite?' can depend on how you define space and time separately," explains Aguirre. "There is spacetime, that's what Einstein teaches us; we can choose to cut it into space and time separately in many different ways. They're all fundamentally valid, they'll all give the same results to any particular experiment we think of, but they have different intellectual implications and some are much more convenient for certain purposes than others."

"If you've got an infinite spacetime, there will often be certain ways that you can cut it up so that it looks like the Universe is, say, finite and expanding. [It may be expanding] forever and getting infinitely big, but at any time it's finite. At the same time, the very same spacetime can be chopped up in such a way that at any time it's spatially infinite, so it's an infinite, expanding Universe." In an inflationary Universe it turns out that once the inflation stops there is a most natural way of chopping it up; a way in which the Universe is close to homogeneous. And this gives a Universe that is spatially infinite. "Inflation very naturally gives rise to homogeneous infinite universes that would evolve into something like what we see. I think it's really neat that we can get suggestive evidence for such a rich, and multifaceted, and interesting picture [in which] the Universe is infinite."

### Infinity, actually

The question of whether the Universe is infinite in extent concerns one type of Aristotle's infinities, potential infinities, which we can imagine but never actually see. But there's another type of infinity that Aristotle talked about, *actual infinity*. Here something localised, something that we can actually measure, becomes infinite. One situation in which such an actual infinity could arise in the Universe is within a black hole, which forms when a massive object like a star, for example, starts collapsing in on itself with nothing stopping it. Theory would suggest that this leads to an infinite density of mass at a single point. Do such infinities exist in the Universe?

Anthony Aguirre. Image: Kelly Castro.

"A black hole is not necessarily a solid object, it's just a sort of surface in the Universe," explains Barrow. "If you go inside it you can't get back out, because you would need to move faster than the speed of light to escape from its gravitational pull. What happens is that a big cloud of stuff collapses and gets denser and denser. Eventually a surface will form around it which we call a horizon. Once you're inside the horizon of a very large black hole that's, say, a billion times the mass of the Sun, the conditions would just be like they are in this room, nothing odd. But if you tried to backtrack and leave, you would find that you couldn't do it. And once you're inside the black hole, things do continue to move towards unlimitedly high density at the centre. If you're on the outside, you can't see any of that; it's hidden from you. Its effects are insulated; they can't affect the outside Universe."

"Long ago, Roger Penrose made a conjecture that's known as *cosmic censorship*. [It says] that if singularities or infinities were to form in the Universe and nothing could stop them, then they would always be trapped within these horizons. There can't be what people call "naked" singularities, so you can't have an infinity that affects us on the outside. In various cases this is proved, but it's far from being proved in general; it's a very difficult mathematical problem."

Another type of infinity that might be living right here amongst us is the infinitely small. Or, put differently, the infinitely divisible. If we had super accurate rulers and pencils, could we keep dividing a line segment into smaller pieces forever and ever, creating pieces that are as small as we like?

Ellis thinks the idea is preposterous. "If you hold your fingers 10cm apart and if you believe that there's a real line of points, like in mathematics, between your fingers, then there's an uncountable infinity of points between your fingers. That's completely unreasonable; I believe that's a mathematical idea which does not correspond to physics. Richard Feynman said that the most important thing he would want to leave to future generations, if he had to just leave one thing to them, was the statement, "Matter is made of atoms." I think we have good reason to believe that there's a similar statement about spacetime, that spacetime is made of atoms. If you hold your fingers that far apart, there's a very large number of physical points but it is not infinite and it's not uncountable."

If spacetime is made out of indivisible pieces then there must be a smallest length scale; a shortest possible length. Physical theories do indeed support this idea, conjecturing that there is nothing shorter than the so-called *Planck length*. It's absolutely tiny, around 10^{-35} metres: that's a number with 34 zeroes after the decimal point. Current measurement instruments cannot come anywhere near resolving something so small, but the idea is that even in theory, even if we had very, very powerful instruments we could never measure anything smaller than a Planck length. (See this *Plus* article to find out more.)

### The cosmic hot dog

Ellis made an important distinction. There is the mathematical concept of infinity on the one hand, which holds, for example that a line is infinitely divisible, and the physical concept on the other, which concerns real quantities and phenomena that may or may not exist in nature. But there is also a third type of infinity, one that we probably feel most familiar with.

"Nowadays we would distinguish between mathematical infinities, physical infinities and *transcendental infinities* which theologians or philosophers might talk about," says Barrow. "Those transcendental infinities are what the average person on the street, if you mention infinity to them, feels quite at home with; they think they know what this is about. It's a sort of cosmic everything: what did the mystic say to the hot dog salesman? 'Make me one with everything.' "

"In many religious traditions, that totality [of everything] might even be the same thing as God or some other cosmic ultimate. That's a different thing to what physicists and mathematicians are trying to deal with, which is more specific. You can look through the history of ideas and mathematics and physics, and it would be an option for people to say, 'I believe in mathematical infinities or not,' or, 'I believe or disbelieve in physical infinities,' or, 'I believe and disbelieve in any other type of transcendental infinity.' You can find every permutation of beliefs and disbeliefs. You've got 2^{3}=8 options."

Infinity: a hot dog with everything? Image: TheCulinaryGeek.

And opinions are indeed divided. Both Barrow and Aguirre are happy with mathematical infinities and don't shut the door on physical ones either. "I think it's certainly true that we can develop theories that have infinity in them that can be perfectly useful," says Aguirre. "It's certainly true that as finite beings we can only experience a finite part of [the Universe], but I can't see any reason to place limits on whether the Universe can be finite or infinite in principle."

Ellis, on the other hand, does not believe that physical infinities exist and points to potential problems with using infinity in mathematical arguments pertaining to physics. He refers to a famous thought experiment due to the mathematician David Hilbert. Suppose you have a hotel with an infinite number of rooms, and suppose that the hotel is full. The paradox is that you can still fit a new person in; you simply move every person in the hotel one room along, so the person from room 1 goes into room 2, the person in room 2 goes into room 3, and so on. Since there isn't a largest number, you can do this without making anyone homeless and then you can fit the new person into room one.

Because of paradoxes like this one, Ellis thinks that you have to be very careful when you use infinities in a physical context. "I'll make a distinction; there are some times when people talk about infinity when all they really mean is a very large number, and they're just using infinity as a code word for a large number. In that case, I think it's more informative to make a guess what that large number is and to talk about that large number, not infinity. There are some cases where people use infinity in its deep sense; in the paradoxical sense. The paradoxical sense is, for instance, Hilbert's hotel. In my opinion, if a physics argument or any other argument depends on those paradoxical arguments, then this is a false argument and it should be replaced by something else."

In summary there is as yet no consensus as to whether infinities exist in the physical world. In the absence of concrete scientific answers it makes sense to turn to philosophers. "I think it's very important to get physicists and philosophers together," says Aguirre. "I think there's a reaction that a lot of my physics colleagues have about philosophers which is that they don't know any physics. They're just saying things about physics and they don't really know what they're talking about, and criticising physics but they don't really understand it. I think there may once have been some truth to that and I'm sure that there is now, but the philosophers that I talk to all know lots of physics. I see them as being specialists in thinking about the intellectual foundations of those questions, looking at them from a slightly bigger and different point of view than a more empirically or pragmatically engaged physicist would. I think that's incredibly valuable."

### About the authors

Marianne Freiberger and Rachel Thomas are editors of *Plus*.

## Comments

## The Law of Conservation in relation to 0

With the law of Conservation of Energy we find that energy can't be created nor destroyed. Therefore the energy that exists today was never created, and will never be destroyed. This energy will only continue to evolve through entropy. The relation I find with this to the concept of 0, depends on the definition of zero. If zero is a term referring to nothing, it can't exist within our physical existence because we can never detect or even account for "nothing" in a reality where everything is something - energy. We also know that even space itself is expanding, and all of the energy we can detect is in motion, giving it a value greater than zero...

When we apply the value of 0 to something in our reality, we remove it from existence. An object with 0 mass has 0 energy and cannot interact with photons, because it has no intrinsic properties - no motion, no interaction with light, no positive, negative, or neutral charge... In other words it doesn't exist.

The concept of 1-1=0 is a manifested concept to create a numerical measurement to something that doesn't exist. If something doesn't exist and we can't measure how much of it is not here then we equate it to the value of zero. The truth of the matter is, no matter how much of something we don't have, we still have something - or a mathematical value of >0

What do you think?

## The Law of Conservation in relation to 0

With the law of Conservation of Energy we find that energy can't be created nor destroyed. Therefore the energy that exists today was never created, and will never be destroyed. This energy will only continue to evolve through entropy. The relation I find with this to the concept of 0, depends on the definition of zero. If zero is a term referring to nothing, it can't exist within our physical existence because we can never detect or even account for "nothing" in a reality where everything is something - energy. We also know that even space itself is expanding, and all of the energy we can detect is in motion, giving it a value greater than zero...

When we apply the value of 0 to something in our reality, we remove it from existence. An object with 0 mass has 0 energy and cannot interact with photons, because it has no intrinsic properties - no motion, no interaction with light, no positive, negative, or neutral charge... In other words it doesn't exist.

The concept of 1-1=0 is a manifested concept to create a numerical measurement to something that doesn't exist. If something doesn't exist and we can't measure how much of it is not here then we equate it to the value of zero. The truth of the matter is, no matter how much of something we don't have, we still have something - or a mathematical value of >0

What do you think?

## Hilbert is just fine with me.

I see no paradox at all. I was delighted when I was back in high school (a very very long time ago ... maybe an infinite time ago), and a completely valid argument under the generally accepted theories of maths at the time. It's a fantastical journey into Cantor and his amazing "rules" for talking about infinity. But that's strictly a mathematical idea, and at that level you can't easily defy it, but you're welcome to offer another theory of your own. (;-{}

And the question of whether the continuum = aleph1? Which is a beautiful question to ponder while on sabbattical. Ah, the fresh air!

It's a mathematical thing more than a physical thing. We don't really ABSOLUTELY know if any sort of infinity exists outside our brains. GregR

## The last number of infinity

What's the last number of infinity?

Since infinity goes on for infinity, there can't be a last number. And if there's no last number, then what's that last number?

Give up?

Well, duh, zero, of course. Zero is the very last number. If there's no last number, then it's zero.

And here you were thinking there was no such thing as a last number. Ya learns somethin' new every day...

A math joke you can share at your next mathematics conference and everyone else will think you're the life of the party telling such funny jokes. Or, they'll groan at how bad it is. It could go either way.

## the universe and the atoms

I think that infinity is the relationship between the universe and the atoms or even smaller particles.

## (almost) infinitely large typo error...

I think the Plank length in metres is 10 to the MINUS 35th power... or 1/(10 followed by 34 zeros)....

## Still wrong?

I think an even more accurate statement is that Plank length in metres is:

[0. (followed by 34 zeroes) (followed by 1)]

A number followed by 34 zeroes is still just the number. (E.g. 0.10000000000... is still just 0.1; but 0.0000000001 is a very small number.)

## Yes, thanks infinitely for

Yes, thanks infinitely for pointing that out!

## Nature of infinity re: integer numbers...

This is a very deep and serious issue. There are some fundamental questions to ask about infinities, which may elucidate some deeper properties.

Take a simple infinity like that of integer number. As is a single entity, it must contain 'All Integer Number'. Note that this is a descriptor of a total unified set of a particular 'quality' of number. In this state of a single entity of the 'infinity of all integer number', viewed from the outside as it were, there is no particular (as in 'particles') structure where individual numbers are visible. This is the non-computational reality of the infinity.

And this is the source of this particular infinity paradox. If ALL integer numbers are there, then the structure is fixed, as a dimension-like [realm of quality of all integer number]. So on the one hand we have a unified infinite realm, a single entity of infinity, where the infinity can only be unified if all interger numbers are viewed as a 'whole'. While on the other viewing its 'contents' as individual particles of number we see individual particulate numbers, but an infinite number of them, which again is paradoxical. Interestingly there seems to be something of a 'wave/particle duality' paradox in this view of infinity.

The Hilbert Hotel makes a different interpretation of the infinity boundary viewed from the inside at the particulate level it assumes that because there is no limit on magnitude you can keep adding numbers infinitely - this is a local view with an indeterminate, infinite, boundary. Whereas the dimension-like view is to regard the infinity of integer number as a 'whole', a 'fixed', non-computational dimension-like unified realm of quality of all integer number.

The description of integer infinity that I gave above creates an interesting view. The infinity of integer number, as a dimension-like entity represents a change of state, a change of phase from the view of particulate individual numbers to a unified whole. This infinity refers to a 'realm of quality' and is a non-calculable entity, whereas the particles of number viewed inside are calculable, yet infinite.

Discuss.