# Infinity

**Infinity** is the *concept* of an object that is larger than any number. When used in the context "...infinitely small," it can also describe an object that is *smaller than* any number. It is important to take special note that infinity is **not** a number; rather, it exists only as an abstract concept. Attempting to treat infinity as a number, without special care, can lead to a number of paradoxes.

Infinity is

nota number!

Infinity is often used in describing the cardinality of a set or other object (such as a list or sequence of terms) that does not have a finite number of elements. Care must be used to avoid confusion, as nonintuitive results often present themselves: for example, the set of integers and the set of even integers have the same size, despite one being contained within another.

Infinity is also used to describe the limiting behavior of some functions, where a function "approaching infinity" means that it grows without bound. For instance, \(f(x) = x^2\) approaches infinity as \(x\) grows large, but \(f(x)=\frac{2x+1}{x+1}\) does not approach infinity as \(x\) grows large (instead it approaches 2, as in the theory of limits).

Infinity is denoted by the symbol \(\infty\).

The concept of infinity is extremely important in a variety of contexts, most notably calculus and set theory. It is also useful in geometry (by analyzing infinitely close points) and inequalities (by analyzing the effect of an infinitely small change), as well as many other areas where the effects of an infinitely small change can be analyzed.

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## Intuitive Explanation and Popular Usage

Though infinity leads to many counterintuitive results, the concept itself is relatively straightforward:

Infinity is not a number, but if it were, it would be the largest number.

Of course, such a largest number does not exist in a strict sense: if some number \(n\) were the largest number, then \(n+1\) would be even larger, leading to a contradiction. Hence infinity is a *concept* rather than a *number*. Put another way, infinity is the concept that there is no largest number. Infinity is used to describe quantities which go on forever without end.

Common visualizations of infinity include

- the length of a line (or ray) that extends forever;
- the amount of time it would take one to constantly travel along a circle;
- the number of digits in an irrational number, most notably \(\pi\);
- the number of points within any boundary, commonly a dartboard.

The term "infinity" is also sometimes misused to refer to an unusually large number, that can be considered infinite for practical purposes. For instance, the number of possible chess positions is often referred to as infinite, though this is not true in a strict sense (in fact, there are roughly \(10^{120}\) such positions). Similarly, the universe is commonly thought of as being infinite, but in fact this is an open question.

## Arithmetic with Infinity

Because infinity is not a number, it requires special rules of arithmetic in order to remain consistent. This is a useful view in fields such as measure theory, which extends the real numbers by adding two numbers \(\infty\) and \(-\infty\), that satisfy some additional rules:

\[\begin{align} a + \infty &= \infty + a \\ &= \infty &&(\text{for any } a \text{ besides } -\infty)\\\\ a - \infty &= -\infty + a \\ &= -\infty &&(\text{for any } a \text{ besides } \infty)\\\\ a \cdot \infty &= \infty \cdot a \\ &= \infty &&(\text{for positive } a)\\\\ a \cdot \infty &= \infty \cdot a \\ &= -\infty &&(\text{for negative } a)\\\\ \frac{a}{\infty} &=\frac{a}{-\infty}\\ &=0 &&(\text{for real } a)\\\\ \frac{\infty}{a} &= \infty &&(\text{for positive } a)\\\\ \frac{\infty}{a} &= -\infty. &&(\text{for negative } a) \end{align}\]

It is worth noting that \(\frac{1}{0}\) is **not** \(\infty\). Additionally, operations involving multiple infinities \((\)such as \(\infty - \infty\) and \(\frac{\infty}{\infty})\) are not generally well-defined.

Importantly, note that this is an *extension* of the real numbers chosen specifically to allow infinity to be treated as a number; without this context, infinity remains a *concept* rather than a number. For instance, the following sections continue to treat infinity as a concept rather than a number, because the measure theory context above no longer applies.

## Infinity as a Limit

Most commonly, the term "infinity" is used to refer to an arbitrarily large number; i.e. a number that grows without bound. Thus, arithmetic involving infinity can be performed, with the convention that \(\infty\) represents a number that is as big as necessary. For instance, though \(\infty \times \infty\) is a meaningless collection of symbols, it can be understood as \(n \times n\) for a very big number \(n\). Formally, such an expression is written as

\[\lim_{n \rightarrow \infty} n^2,\]

which is read as "the limit as \(n\) goes to \(\infty\) of \(n^2\)." This is itself larger than any other number, so it can be written that \(\lim_{n \rightarrow \infty} n^2 = \infty\). Limits can also have finite results; for instance, \(\lim_{n \rightarrow \infty} \frac{1}{n} = 0\).

This concept is useful when comparing how "quickly" expressions *tend to infinity*, or (informally speaking) get larger. For instance, though \(2 \times \infty\) and \(\infty\) represent essentially the same concept (both these "numbers" are larger than any other), the former seems as though it should be larger in some sense, and indeed (under the correct interpretation), this is the case. More formally,

\[\lim_{n \rightarrow \infty} (2 \cdot n) = \infty, \quad \lim_{n \rightarrow \infty}(n) = \infty\]

but

\[\frac{\lim_{n \rightarrow \infty} (2 \cdot n)}{\lim_{n \rightarrow \infty}(n)} = \lim_{n \rightarrow \infty} \frac{2 \cdot n}{n} = \lim_{n \rightarrow \infty} 2 = 2,\]

so the former expression is "twice" the second.

Note that this also demonstrates one pitfall of considering \(\infty\) as a number: the above example shows that \(\frac{\infty}{\infty}\) can be 2, even though it seems like it should be 1. There are several other such arithmetic functions that result in seemingly illogical results, the so-called indeterminate forms:

\[\begin{array} &\frac{0}{0}, &\frac{\infty}{\infty}, &0 \cdot \infty, &0^0, &\infty^0, &1^{\infty}, &\infty-\infty, \end{array}\]

which can all take on any value for suitably chosen functions (the case of \(\frac{\infty}{\infty}\) is demonstrated above).

These considerations, in which functions are evaluated at infinity, form the basis for calculus.

\[\large \lim_{x \to 0} x^0 , \quad \lim_{x \to 0^{+}} 0^x \ , \quad \lim_{x \to 0^{+}} x^x,\quad \lim_{x \to \infty} 0^x \]

Let \(A,B,C,D\) denote the values of the 4 limits above (in that order), respectively.

Find the values of these 4 limits. Submit your answer as the 4-digit integer \(\overline{ABCD}\).

## Infinity in terms of Cardinality

The **cardinality** of a set is the number of elements it contains. For example, the sets \(\{1,2,3,4,5,6\}\), \(\{\pi, \pi^2, e, e^2, i, -1\}\), and \(\{\color{red}{\text{red}}, \color{orange}{\text{orange}}, \color{yellow}{\text{yellow}}, \color{green}{\text{green}}, \color{blue}{\text{blue}}, \color{purple}{\text{purple}}\}\) all have cardinality 6. One way to show this is to simply count the elements in each set, but this can be difficult for larger sets (and impossible for infinite ones). In general, the cardinality of sets can more easily be found using **one-to-one correspondences** (also called bijections), which essentially means pairing up the elements of two sets:

Set 1 | Set 2 |

1 | \(\color{red}{\text{red}}\) |

2 | \(\color{orange}{\text{orange}}\) |

3 | \(\color{yellow}{\text{yellow}}\) |

4 | \(\color{green}{\text{green}}\) |

5 | \(\color{blue}{\text{blue}}\) |

6 | \(\color{purple}{\text{purple}}\) |

which shows that this set of colors has cardinality 6. This approach would be useful, for instance, in quickly finding the cardinality of the set \(\{10, 20, 30, 40, 50, 60\}\).

A set with an infinite number of elements has infinite cardinality. For instance, the set of positive integers

\[\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \ldots\}\]

has infinite cardinality. Similarly, the set of the rational numbers also has infinite cardinality, as does the set of the real numbers, the set of even numbers, and so on.

Intuitively speaking, it would seem that the set of natural numbers has more elements than the set of even natural numbers, since the former set contains the latter:

\[\{2, 4, 6, 8, \ldots\} \subset \{1, 2, 3, 4, 5, 6, 7, 8, \ldots\},\]

but in fact both these sets have the *same* cardinality. This can be seen using the one-to-one correspondence method from above (since counting the number of elements in each set is impossible):

Natural numbers | Even numbers |

1 | 2 |

2 | 4 |

3 | 6 |

4 | 8 |

\(\vdots\) | \(\vdots\) |

In fact, it can also be shown that the set of rational numbers has the same cardinality as the set of natural numbers, but the set of real numbers has *larger* cardinality than the set of natural numbers.

This implies a rather counterintuitive result:

There are

several different kindsof infinities.

This is especially important in set theory and the formalization of mathematics, such as the continuum hypothesis.

## Paradoxes involving Infinity

Though the concept of infinity appears straightforward, its implications can lead to highly counterintuitive results. The most famous of these are **Zeno's paradoxes**, of which there are three major ones:

**1. Achilles and the tortoise**

Achilles (a very fast warrior) and a Tortoise are having a race. Because Achilles is much faster than the tortoise, he has given the tortoise a head start of 100 meters. After some time, Achilles has run 100 meters, while the tortoise has run a much smaller distance of 10 meters. But then after Achilles runs the next 10 meters, the tortoise has completed an additional meter. By the time Achilles reaches that point, the tortoise will have moved forward once again, and this occurs each time Achilles reached the previous position of the tortoise. Thus Achilles can never catch up to the tortoise.

**2. Dichotomy paradox**

Suppose Achilles were instead running to a fixed point (e.g. the finish line). First, he must travel half the distance to the point. Then he will travel half of the remaining distance, then half of that remaining distance, and so on. But then he must complete an infinite number of steps to reach the finish line, so Achilles will never reach the final destination.

**3. Arrow paradox**

Consider an arrow in flight. At any instance in time, the arrow is at some fixed position, and it cannot be moving:

- to a different point, because no time has elapsed for it to move,
- to the point at which it is, because it is already there,

hence no motion occurs at any time, and the arrow cannot complete any movement at all.

All three of these conclusions are clearly absurd (as movement occurs on a regular basis every day), but the flaw in Zeno's reasoning is not immediately obvious. In fact, resolving these paradoxes requires a much more robust framework of propositional logic and calculus, particularly the formal \(\epsilon-\delta\) definition of limits.

Additional paradoxes involving infinity include Russell's paradox and **Galileo's paradox**, which notes that

There are the same number of squares as there are numbers, though most numbers are not squares.

This (in modern language) is a statement about cardinality from the previous section.

Consider the three sets

- \(A=\left\{ x\in \mathbb R|\sin(x)=-1 \right\}\)
- \(B=\left\{ x\in \mathbb R|\sin(x)=0 \right\}\)
- \(C=\left\{ x\in \mathbb R|\sin(x)=1 \right\}.\)

Which of them has the largest cardinality?