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Understanding the behaviour of infinity is one of the major accomplishments of mathematics. Sadly, the infinite is often misunderstood and could lead to various paradoxes when used or interpreted the wrong way. This FAQ attempts to explain the role of infinity in mathematics and tries to resolve a few apparent paradoxes.

Infinity is not a real number!
Very often, people try to work with infinity like they work with ordinary real numbers. They soon end up with paradoxical situations, like
It’s certainly true that $2\infty=\infty$. Divide both sides by infinity and we get that 2=1.
This is an absurd result.The resolution to this apparent paradox is simply that infinity is not a member of the set of real numbers. We can often adjoin infinite quantities to $\mathbb{R}$, but these infinite things do not behave like ordinary real numbers and not every calculation with infinite things is allowed. For example $\frac{\infty}{\infty}$ will often not be allowed!!

When asked to solve a problem in the set of real numbers, then working with infinity is very often not allowed (because again: infinity is not a real number). If you do want to work with it, then you need to mention this explicitely and you need to be very careful about the rules you use!

Why bother with infinity if infinity doesn’t exist in reality?
Firstly, we don’t know that. But actually it’s irrelevant whether infinity exists in reality. Mathematicians work with infinity because it’s easier than not working with infinity.

Say, for example, that you take a group of people and you want to measure their height. You may obtain answers like 1.70m, 1.76m, 1.84m, etc. However, you want to make a mathematical model for the experiment. It is now convenient to say that the space which contains all the lengths is just $\mathbb{R}$. Of course, you will never encounter a length like $\sqrt{\pi}$ meters, but that’s not important. Choosing $\mathbb{R}$ was easy and convenient.

Furthermore, if we choose $\mathbb{R}$, then we can apply the mighty methods of calculus on our outcomes. For example, we may find a curve that fits our outcomes best and we can find the slope and area under that curve. If we didn’t choose our space infinite, then this was impossible (or very difficult at best)!

So, what is infinity?
The first thing that we must understand is that there is no unique definition of “infinity” in mathematics. There are very different kinds and flavours of infinity. All these different interpretations of infinity have their purpose.

Sometimes, infinity is just a symbol, this happens for example

In limits: with notations such as $\lim_{x\rightarrow +\infty}{f(x)}$, or $\lim_{x\rightarrow a}{f(x)}=+\infty$.
The order of an element in a group.

However, while the above are simply symbols, it is often very useful to give them some kind of meaning anyway. We do this by adjoining some infinite quantities to our original set. This way, the $\infty$-notation in limits becomes an actual limit. Furthermore, we are often allowed to do all kinds of arithmetical operations on the infinite quantities.

The extended real line is $\mathbb{R}\cup \{+\infty,-\infty\}$.
The projective real line is $\mathbb{R}\cup \{\infty\}$.
The Riemann sphere is $\mathbb{C}\cup \{\infty\}$.
In nonstandard analysis, there are infinite numbers and infinitesimal numbers.

And finally, infinite can be both. It can be both denoting that something is very big, and at the same time it can be worked with.

The cardinal numbers are used to see how big an infinite set is.

Let’s pick one of each of these lists and let’s explain them a bit further.

How is infinity used in limits?
In limits, infinity is just a notation. For example, the notation $\lim_{x\rightarrow +\infty}{f(x)}=a$ means “if we take x to be really big, then f(x) will be very close to a. If we take x to be even bigger, then f(x) will be even closer to a”. For example, consider the function $f(x)=\frac{1}{x}$. If we take x=1000, then f(x)=0.001 is very close to 0. If we take x=100000, then f(x)=0.00001 is even closer. We say that f(x) converges to 0.

What we’ve just done can be formalized as follows: we say that $\lim_{x\rightarrow +\infty}{f(x)}=a$ ifwe can make the distance between f(x) and a to be as small as desired by taking x to big enough.

As you see, we haven’t really done anything with infinity here. The $\infty$ notation just meant that our numbers can be arbitrarily large (but still finite).

What is the extended real line?
The extended real line was invented because mathematicians weren’t really satisfied with limits. As you saw, limits just denoted that something gets arbitrarily close to a number a if x is chosen very big. However, it kind of makes sense to say that “if $x=+\infty$, then f(x)=a”. In the real numbers, this doesn’t make sense at all, however. Because infinity is NOT a real number. That’s why we adjoin two new elements to $\mathbb{R}$, namely $+\infty$ and $-\infty$. Now it does make sense to say that $f(a)=+\infty$.

We can even do arithmetic with infinity! For example:

$+\infty+\infty=+\infty,~~\frac{1}{+\infty}=0,~~2<+\infty$

However, not all things make sense. For example $+\infty-\infty$ is undefined. See http://en.wikipedia.org/wiki/Extended_real_number_line for more information.
The projective line uses the same idea. Here we adjoin an element $\infty$ to $\mathbb{R}$. The idea here is that $\infty$ encapsulates both negative and positive infinity. So the real line becomes circular: going to the right will get you to $\infty$, but going to the left will also get you there. With this notion of infinity, we can even define things like

$$\frac{1}{0}=\infty$$

We couldn’t do this in the extended real line because the answer could be positive or negative infinity. However, $\infty-\infty$ is still undefined, and we can’t say now that $2<\infty$. See http://en.wikipedia.org/wiki/Real_projective_line for more information.

What are cardinal numbers?
Cardinal numbers are used to distinguish between different sizes of infinity. Some infinite sets are bigger than other infinite sets in some sense.

To illustrate the idea, let’s first look at finite collections. Let’s say we have a really smart toddler, and let’s name him Greg :biggrin:. You then give Greg two sets of marbles and you ask “do these sets contain an equal number of marbles”? Greg can’t count the marbles (he’s too young), but he can use another method to determine whether the two sets are equal. Greg can simply pick a marble from the first set and a marble from the second set and put them together. He keeps doing that same thing until all the marbles are alligned with other marbles. If there are left-over marbles that cannot be assigned to another marble, then Greg can decide that one set was bigger than another.

The same thing happens with infinite sets. We can’t really count the infinite sets, but when we’re given sets A and B then we can easily say when those two sets have an equal number of elements:
We say that A and B have the same cardinality if there exists a one-to-one correspondence between A and B.
This looks like a plausible definition, and if you think about it: it’s about the only thing we can do for arbitrary sets! However… there’s a catch. Consider the sets

$$A=\{0,1,2,3,4,5,…\}~~\text{and}~~B=\{0,2,4,6,…\}$$

You could argue that A is definitely bigger than B because B is contained in A. This is not true, however! We can find the following one-to-one correspondence

$$A\rightarrow B:n\rightarrow 2n$$

Every element in A is now assigned to an element of B and conversely. So A and B are the same size! This is a paradoxical situation, but we just got to get used to it. It’s the nature of infinity.

Using some clever reasoning, we can even say that the set of natural numbers, the set of integers and the set of rational numbers all have the same number of elements. The set of real numbers is much larger though! Cantor was the first to discover these things.
The following forum members have contributed to this FAQ:
bcrowell
DaveC426913
Hurkyl
micromass
PAllen
Redbelly98

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8 replies
1. mrnike992 says:

Semi-relevant and interesting problem I stumbled across the other day: If you have an infinite number of infinitely small objects, would it take up near-zero volume or infinite volume?

2. Hamza Abbasi says:

(Y) Wonderful !! I have a question if infinity doesn't fall in the set of real number than ever do it stand ?? Umm complex no?

3. jeffery_winkle says:

The observable universe is a sphere, centered on us, where the radius is the horizon distance, which is the distance light could have traveled since the Big Bang. The observable universe is finite, but is only a subset of the entire universe. Is the entire universe finite or infinite? If the universe has positive curvature, like a sphere, it could be finite. If it has zero curvature, like a plane, or negative curvature, like a hyperboloid, it is infinite. According to all our measurements, the universe is flat, and thus infinite. There is still the possibility that the universe could have positive curvature, and be finite, but where the radius of curvature is so large, that the deviation of flatness, from our point of view, that it would appear flat to us.It used to be thought that the universe might have enough mass to recollapse into a Big Crunch. This has been disproven. In fact, the expansion of the universe is accelerating. That means, we know that it will exist for an infinite length of time in the future.The Big Bang has been confirmed by the CMB, so we think of the Big Bang theory as having won the Big Bang versus Steady State debate. However, we still don't know whether the Big Bang was the fundamental beginning of time, which is the traditional view, was instead only a local Big Bang, which created this specific part of the universe, which we think of as the universe. According to chaotic inflation, at any time, a given patch of space might suddenly undergo inflation. According to this view, time would extend infinitely backwards.So does infinity exist in the real universe? According to our recent theories, the universe is very probably infinite in space, definitely infinite in future time, and possibly infinite in past time. There are other occurrences of infinity in physics, such as having to sum over an infinite number of Feynman diagrams.Someone here acted like if you can't count to infinity, then infinity doesn't exist. That is a misunderstanding of infinity. When you ask, "Can you count to such and such number?", what you are asking is, does number X appear in the set of integers, Z = 1, 2, 3, …? Well, the number 1/2 also does not appear in the set of integers. Does that mean 1/2 does not exist? Why single out the integers as your set of comparison? Why not choose some other set, such as the prime numbers? Why not say the number 9 does not exist because it does not appear among the prime numbers? You can't count to infinity. You also can't count all the numbers that appear between 0 and 1. Does that imply that these numbers don't exist? You can't write down all of the digits of pi. Does that imply pi doesn't exist?

4. jeffery_winkle says:

Even elementary introductory physics requires at least simple calculus which necessarily involves the concept of infinity.

5. Umbrazno says:

You can experience infinity in real life. Try to imagine NOT existing at all. The closer you get to an authentic visualization, the farther you get also. Try it. Another example is counting. They say "Count as high as you can and then I'll just add one." and "If you find the edge of the universe and then stand on it and shoot an arrow…" The most famous argument against infinity is actually the most sound proof that infinity does exist in the real world. Infinity is an Anti-number of sorts.  The fact that you can "add one" or "shoot another arrow" means that there is actually and anti-force which accommodates possibility. And you want infinitesimal? Add a drop of water to a swimming pool at rest. Does it rise?