Infinity in Mathematics: Limits and Cardinality FAQ

Introduction

Understanding the behavior of infinity is one of the major accomplishments of mathematics. However, the infinite is often misunderstood and can lead to apparent paradoxes when misused or misinterpreted.

This FAQ explains the role of infinity in mathematics and attempts to resolve several apparent paradoxes.

1. Infinity is not a real number!

Very often, people try to work with infinity as 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, working with infinity is usually not allowed (because, again, infinity is not a real number). If you do want to work with it, you need to state that explicitly and be careful about the rules you use.

2. Why bother with infinity if infinity doesn’t exist in reality?

We do not know whether actual physical infinity exists. In any case, it is irrelevant to mathematics: mathematicians work with infinity because it is often more convenient than not working with it.

For example, suppose you measure the heights of a group of people and obtain values such as 1.70 m, 1.76 m, 1.84 m, and so on. To model the experiment it is convenient to regard the space of possible lengths as $\mathbb{R}$. You will never encounter a length exactly equal to $\sqrt{\pi}$ metres in practice, but choosing $\mathbb{R}$ makes the mathematics simpler and more powerful.

With $\mathbb{R}$ available, we can apply the methods of calculus: fit curves to data, compute slopes and areas, and so on. Without an infinite model these tools are either impossible or much harder to use.

3. So, what is infinity?

There is no single definition of “infinity” in mathematics. There are several different notions and each serves a particular purpose.

Sometimes infinity is just a symbol. For example:

• In limits: 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 (sometimes written as “infinite”).

Although the examples above are often symbolic, it is useful in many contexts to give these symbols a precise meaning by adjoining infinite quantities to the original set. In that way the [itex]\infty[/itex>-notation in limits becomes an actual limit and we can sometimes perform arithmetic with infinite quantities within a chosen framework.

• 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.

Finally, “infinite” can serve both roles: it can denote a notion of being arbitrarily large and, in appropriate frameworks, it can denote objects that can be manipulated.

The cardinal numbers measure the size of infinite sets.

Below we pick a few of these notions and explain them further.

4. How is infinity used in limits?

In limits, infinity is often just a shorthand. For example, the notation $\lim_{x\rightarrow +\infty}{f(x)}=a$ means “if we take x sufficiently large, then f(x) will be arbitrarily close to a; taking x larger still makes f(x) closer to a.” For instance, with $f(x)=\frac{1}{x}$, choosing x = 1000 gives f(x) = 0.001 (close to 0); choosing x = 100000 gives f(x) = 0.00001 (even closer). We say that f(x) converges to 0.

This notion can be formalized: we say $\lim_{x\rightarrow +\infty}{f(x)}=a$ if, by taking x sufficiently large, we can make the distance between f(x) and a as small as desired.

As you can see, the symbol $\infty$ here indicates that values can be arbitrarily large but still finite.

What is the extended real line?

The extended real line was introduced to make certain limit statements more convenient. In the real numbers the expression “x = +\infty” does not make sense because infinity is not a real number. By adjoining two new points, $+\infty$ and $-\infty$, to $\mathbb{R}$ we obtain the extended real line, and expressions such as $f(x)=+\infty$ become meaningful in that enlarged set.

Within the extended real line some arithmetic expressions are defined, for example:

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

However, not all expressions are defined there: for instance $+\infty-\infty$ is undefined. See Wikipedia: Extended real number line for more details.

The projective line uses a different idea: it adjoins a single element $\infty$ to $\mathbb{R}$ that represents both directions at once, so the real line becomes topologically a circle. In that model one may write

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

but $\infty-\infty$ remains undefined, and one cannot in general compare ordinary real numbers with the projective $\infty$ (so statements such as $2<\infty$ do not make sense there). See Wikipedia: Real projective line for more information.

5. What are cardinal numbers?

Cardinal numbers distinguish different sizes of infinity: some infinite sets are, in a precise sense, larger than others.

To see the idea, first consider finite collections. Suppose a toddler named Greg is given two sets of marbles and asked whether the sets contain an equal number of marbles. If Greg cannot count, he can still pair marbles one-to-one: pick one marble from the first set and one from the second set and match them until no more pairings are possible. If one set has unpaired marbles remaining, that set is larger.

The same notion extends to arbitrary sets. We say that sets A and B have the same cardinality if there exists a one-to-one correspondence between A and B.

There is a counterintuitive consequence for infinite sets. Consider

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

Although B is a subset of A, A and B have the same cardinality via the bijection

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

Every element of A is paired with a unique element of B, and conversely, so A and B are the same size. This is a standard phenomenon in infinite sets.

Using similar reasoning one shows that the set of natural numbers, the set of integers and the set of rational numbers are countably infinite and therefore have the same cardinality. By contrast, the set of real numbers is uncountable and strictly larger. Cantor was the first to demonstrate these differences.

Contributors to this FAQ:

• bcrowell
• DaveC426913
• Hurkyl
• micromass
• PAllen
• Redbelly98