Can anything 'real' be infinite?

We often here the terms 'infinite' and 'infinity', sometimes used in connection with the size of the universe, such as an infinite universe for example. The terms more properly belong in the world of mathematics, where for example we may have an infinitely long string of numbers as the result of a calculation, such as pi. The terms are very real to mathematics, but can anything real, not theoretical, be infinite?

An infinite Universe?

Coma galaxy cluster

Naturally in mathematics we can have infinity, numbers go on for ever, but numbers are not real, they are abstract. I cannot imagine anything 'real' that we could apply an infinite number to. The only thing I can imagine that could be really infinite is nothing, the 'nothing' I described earlier in Where did the universe come from? and we have no idea if that exists.

The concept of infinity is a puzzling one. For example: imagine a standard pack of playing cards that consists of just one of each card but two jokers. Imagine that the packs of playing cards are infinite in number (A thought exercise only of course). We therefore have more jokers than any other card in each pack, so do we have more jokers in total? You could reply that as the packs are infinite in number they can't be counted so it would be impossible to know. However, as the ratio of jokers to other cards in each pack is fixed, then at any number of packs there will always be more jokers. This would appear to indicate, that mathematically, we can have degrees of infinity. Sounds odd doesn't it? It is a valid mathematical argument though.

We have a theory for black holes that describes infinite density. See Exploding Black Holes? What does it mean, other than an unresolvable equation that occurs in mathematics? Exactly what is infinite density? Taking a rather simplistic view it could be argued that if one black hole has infinite density then nothing else can have ANY density. Clearly though in this sense we can have lots of infinite density, so the term obviously carries a meaning in mathematics that does not have the same meaning outside of it. Is the term used in the theory only because that is the way the sums work out, regardless as to its significance in the real world, or is it real?

Strictly speaking, according to Einstein's Theory of Relativity, a singularity does not contain anything that is actually infinite, only things that MOVE MATHEMATICALLY TOWARDS infinity. A black hole is formed when large stars collapse and their mass has been compressed down to a very small size and the powerful gravitational field so formed prevents anything, even light, from escaping from it. A black hole therefore forms a singularity at its centre from the concentrated mass of the collapsed star itself and from the accumulated mass that is sucked into it. A singularity's mass is therefore finite, the 'infinity' refers only to the maths.

Can we have an infinite universe for example? The answer is no, the universe is finite. Stephen Hawking in 'A Brief History of Time' (1989 page 44) describes the universe as being "finite but unbounded". The simplest answer is that as the universe is known to be expanding, it cannot possibly be infinite. To be precise, the dictionary definition of the word universe is "all that is. The whole system of things." In this sense the universe is not expanding into anything other than itself, for whatever it is expanding into is part of the universe, there being nothing else but the universe. However, for the sake of simplicity, I am referring only to our Big Bang expanding universe as 'the universe'. (Even if you happen to disagree with the Big Bang theory, the term 'universe' will still have the same meaning here, as it refers to 'our' universe only, and does not include whatever may or may not exist outside of it.) I will try and explain a finite universe as some people understandably have problems with it.

A good place to start is to understand the very real difference between infinity and a large number.

For example, imagine an ordinary size diamond, as you would expect to find set in a typical lady's engagement ring. Now imagine a super-being armed with super-tweezers, picking out atoms from this diamond one at a time, one every second, since the creation of the universe, some 13 billion years ago. How much of the diamond would by now have been removed? The answer is you couldn't tell without looking through an electron microscope, less than a millionth of the atoms would have been removed. Try and imagine how many atoms there are in that diamond. Now try and imagine how many atoms there are in the entire universe. It is a very large number, but it is finite, and is 10 followed by 80 zeros, (maybe a few more zeros, maybe a few less), expressed as 10 to the 80th. If you want to see what it looks like.........

100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

0r written as - One hundred million, billion, billion, billion, billion, billion, billion. billion, billion

Even this very large number would count as nothing when compared with infinity, because infinity is NOT A LARGE NUMBER be absolutely clear on this point, IT IS NOT A LARGE NUMBER, infinity is ALL THERE IS, it is NOT a number. You could keep counting (or measuring) for ever, and never reach infinity, it is only a description. Infinity describes a thing as having no end, no limit, no boundary or edge, it literally goes on FOREVER, ad infinitum.

Because infinity is not a number, large numbers are no 'nearer' to infinity than small numbers. Number 1 billion for example is no nearer to infinity than number 1, because the two, numbers and infinity, are in no way related. It is then impossible to approach infinity, a thing is either infinite and immeasurable, or finite and measurable, it cannot be part way towards infinity. Imagine running up a 'down' escalator, never moving forward. If you run for a week you are no nearer reaching the end of the escalator than if you run for a minute, you cannot get any closer to something that has no end.

An infinite universe for example would exist in every direction forever, there could be nothing else, ONLY the universe. It is then very easy to understand why our universe cannot be infinite, it is because it is expanding. It cannot be both infinite and expanding. It could be infinite OR expanding, but CANNOT possibly be both, that is a contradiction in terms, and we do know it is expanding. For an explanation of the Big Bang and why we know the universe is expanding. See The Big Bang Theory

I understand that many people have a problem with the idea of our universe being finite, that it has an 'end' to it, a boundary. They ask what this boundary would be physically like, as though it were some form of partition that we couldn't get through. However, there is not a particular direction that we could set off in our warp speed space craft that would lead us to a boundary, no matter how far or fast we travelled. The explanation for this seeming impossibility is that space-time is curved, thus you would be travelling in a circle that only appears to be a straight line. If it were possible to direct a laser beam from here through the centre of the universe it would not hit the other side of the universe, it would eventually hit the back of your head (metaphorically speaking). Einstein demonstrated how matter in the universe distorts the space-time continuum by accurately predicting how much our Sun distorted local space. He used a total eclipse of the Sun (as the only time that stars and the Sun can be seen at the same time in close proximity) to demonstrate that a star that was behind the Sun, and therefore not visible from our line of sight, would in fact be visible (in the darkness of totality) because the Sun warps the space-time around it and thus curves the light beam around the Sun, enabling us to see the star. Strictly speaking, the Sun does not actually curve the light around itself, the entire space-time continuum is curved, the light is still travelling in a 'straight' line.

Galaxies naturally create even larger distortions, and the total mass of the universe gives a distortion that results in our 'straight' line of light curving forever through the universe and never reaching the 'end'. That's why Hawking's describes it as 'finite but unbounded'. As an aid to visualization, but not an accurate representation, consider an ant crawling around a huge beach ball and never coming to the end, it would consider the beach ball as infinite as it has no boundaries. If you now consider the ant as only a two dimensional creature and crawling round a three dimensional beach ball, you could understand why the ant would consider the beach ball to be infinite, the three dimensional picture, that shows how restricted its movement really is, is simply not available to it. Thus with the universe, from our perspective, restricted to our view from within the universe, it appears to be infinite, but this is just an illusion, we are confined to the limits of our universe and cannot escape from it. We are bounded within a finite universe.

The next question that people naturally follow up with is to ask what it is that our finite universe is expanding into. This is a good question and one that can never be answered, we will never be able to escape the confines of our universe to find out. We can only theorise about this, and there are plenty of theories to choose from. I tend to think that we are expanding into an infinite nothing, but for a fuller description check out my page Where did the universe come from? but the truth will never be known. Your guess is as good as mine, probably ;-)

So what is there within our universe that we can truly apply the term infinity to? The universe itself is finite. Infinite mass, in black holes for example, would only appear to be a mathematical description. The age of the universe is finite, and even the number of particles in it is finite.

What do I think?

I think infinity exists only as a means of description, such as found in mathematics for example, or any other thing that exists only in the abstract. I do not believe that it has any real existence in the universe such as infinite mass or infinite size. The word 'infinity' is a descriptive term and not a measure of size, and I therefore do not see how it can be applied to anything 'real', as real things can be measured.

I have come across web sites, and maybe you have also, that claim that atoms can be subdivided down into infinity, and that they contain tiny universes within them, and no doubt tiny people as well. Although science has not yet been able to prove we have reached the ultimate elementary particle from which all complexity is built, there is very strong theoretical and experimental evidence to show that quarks could be it. Smaller than quarks enters the realm of energy, not particles, as in string theory. As matter has been subdivided down from complex objects, to parts of the whole, to molecules, to atoms, to particles, to quarks, at each stage we see a simpler model, each stage is less complex than the previous level. All of which is in perfect agreement with the Big Bang model that describes how all matter is built up from simple to more complex elements, stage by stage. So when breaking down complex objects into smaller parts, it would come as a bit of a surprise if suddenly an entire universe popped up at even smaller scales than wave energy. Entire universes tend to be a bit complex!

However, if string theory is shown to be correct, then tiny loops, or strings, of vibrating wave energy may be the smallest, but they are not particles anyway, and strictly speaking quarks aren't either, as they can not exist independently outside of a particle.

Stephen Hawking in his latest book "The Universe in a Nutshell" (2001. page 176) describes the smallest possible size in our universe, termed the Planck length after Max Planck, as being in the region of a millimeter divided by a hundred thousand billion billion billion. If we were able to probe to even smaller sizes, (which is not feasible as it would require particles that reside in black holes) we would not find anything smaller, we would, according to M-theory, see the other 6 or 7 dimensions of the 10 or 11 dimensions that go to make up our universe, only 4 of which are currently observed by us.(3D + time). These extra dimensions of our universe are curled up so small that we are unable to detect them, but the mathematics of theoretical physics have long said they must exist as part of the fabric of our universe. Please note that they are not 'other dimensions' with other universes, merely the smallest possible unit of our universe.

I think we can quite safely rule out infinite smallness.

Some claim that our universe is but an atom of another 'mega-universe', thus giving an infinite expansion in size. If our universe is finite (as indeed it is believed to be) then anything at all may be postulated as to what it may be that it exists in and is expanding into. You could for example propose that our universe is indeed an atom of a 'mega-universe' but equally you could propose that it as an atom of a mega-donkey's hind leg, but as neither hypotheses is testable or falsifiable there is little point in proposing them. That's why this argument regarding infinity is restricted to just our known universe. However, although the idea that our universe may be a single atom within a 'mega-universe' may have some appeal, it is rather fanciful, and our universe bears absolutely no resemblance whatsoever to the atoms that we observe, or indeed to the particles within those atoms. Sorry, sounds fun but just doesn't match up with observation.

I do not believe that infinity exists in our universe.

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Why can't we divide by zero?

Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

Why can't we divide by zero?

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The reason that the result of a division by zero is undefined is the fact that any attempt at a definition leads to a contradiction.

To begin with, how do we define division? The ratio r of two numbers a and b:

r=a/b

is that number r that satisfies

a=r*b.

Well, if b=0, i.e., we are trying to divide by zero, we have to find a number r such that

r*0=a. (1)Butr*0=0

for all numbers r, and so unless a=0 there is no solution of equation (1).

Now you could say that r=infinity satisfies (1). That's a common way of putting things, but what's infinity? It is not a number! Why not? Because if we treated it like a number we'd run into contradictions. Ask for example what we obtain when adding a number to infinity. The common perception is that infinity plus any number is still infinity. If that's so, then

infinity = infinity+1 = infinity + 2

which would imply that 1 equals 2 if infinity was a number. That in turn would imply that all integers are equal, for example, and our whole number system would collapse.

What about 0/0?

I said above that we can't solve the equation (1) unless a=0. So, in that case, what does it mean to divide by zero?

Again, we run into contradictions if we attempt to assign any number to 0/0.

Let's call the result of 0/0, z, if it made sense. z would have to satisfy

z*0=0. (2)

That's OK as far as it goes, any number z satisfies that equation. But it means that the result of 0/0 could be anything. We could argue that it's 1, or 2, and again we have a contradiction since 1 does not equal2.

But perhaps there is a number z satisfying (2) that's somehow special and we just have not identified it? So here is a slightly more subtle approach. Division is a continuous process. Suppose b and c are both non-zero. Then, in a sense that can be made precise. the ratios a/b and a/c will be close if b and c are close. A similar statement applies to the numerator of a ratio (except that it may be zero.)

So now assume that 0/0 has some meaningful numerical value (whatever it may be - we don't know yet), and consider a situation where both a and b in the ratio a/b become smaller and smaller. As they do the ratio should become closer and closer to the unknown value of 0/0.

There are many ways in which we can choose a and b and let them become smaller. For example, suppose that a=b throughout the process. For example, we might pick

a=b = 1, 1/2, 1/3, 1/4, ....

Since

a=b,

for all choices of a we get the ratio 1 every time! This suggests that 0/0 should equal 1. But we could just as well pick

b = 1, 1/2, 1/3, 1/4, ....

and let a be twice as large as b. Then the ratio is always 2! So 0/0 should equal 2. But we just said it should equal 1! In fact, by letting a be r times as large as b we could get any ratio r we please!

So again we run into contradictions, and therefore we are compelled to

let 0/0 be undefined.

It's a common strategy in teaching to simplify concepts when they are first encountered. In other words, it's common for your teacher to lie to you. I just did! Actually, there is a way to make sense of the expression 0/0. The basic idea is to let both the numerator and the denominator become smaller and smaller, and to make the value of 0/0 dependent upon the way in which numerator and denominator approach 0. This is explained more thoroughly here.

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Fine print, your comments, more links, Peter Alfeld, PA1UM

[17-Feb-1997]