Types of infinity
Infinity appears in several different senses: as the size of a set, as unboundedness, and as the limit of a divergent sequence. An infinite set has infinitely many elements, but it can be bounded. Infinite sets come in different sizes: countable and uncountable. A divergent sequence does not converge, but it doesn’t necessarily go to infinity.
Definitions
Missing definition: infinity
A list of items is infinite if it never runs out: no matter how many elements you list, there are more.
Consider a set $S$. The set $S$ is infinite if the number of elements in $S$ is not any natural number.
Consider a set $S$, and assume the axiom of choice. The set $S$ is infinite exactly when $S$ is in bijection with a proper subset of itself (Dedekind infinite).
Examples
-
$[0, 1]$
$[0, 1]$ is uncountable, and so infinite, because Cantor’s diagonal argument shows it cannot be listed.
- The integers $\mathbb{Z}$
-
$\nats$
$n \mapsto 2n$ is a bijection from $\nats$ onto the even numbers, a proper subset.
- $\mathbb{Q} \cap [0, 1]$
- The set of all subsets of $\nats$
A set of numbers is bounded if it does not go on forever in either direction: there is some number that every number in the set is smaller than in size.
A set $S$ of real numbers is bounded if there is a number $M$ such that $\abs{x} \leq M$ for every $x \in S$.
A subset of a metric space is bounded if it is contained in some ball. A sequence or function is bounded if its set of values is bounded.
Examples
-
$[0, 1]$
$[0, 1]$ is compact because it is closed and bounded.
-
$(0, 1)$
$(0, 1)$ is bounded.
- $\mathbb{Q} \cap [0, 1]$
-
$\setdef{1, 2, 3}$
$\setdef{1, 2, 3}$ is a finite set, so it is bounded and compact, as is every finite subset of $\reals$.
Non-examples
-
The sequence $1, 2, 3, \dots$
The sequence $1, 2, 3, \dots$ is not bounded.
-
$\nats$
$\nats$ is not bounded, because for every $N > 0$, we can pick $n \in \nats$ such that $n > N$.
Missing definition: divergent
Missing definition: countably_infinite
Consider an infinite set $S$. The set is uncountable if it is impossible to list the elements of $S$ in an (infinite) list $a_1, a_2, a_3, \dots$. The set is so large that every such list must leave out some elements of $S$.
Consider an infinite set $S$. The set $S$ is uncountable if there does not exist a surjection from $\nats$ onto $S$.
Examples
-
$[0, 1]$
$[0, 1]$ is uncountable because Cantor’s diagonal argument shows that no list of numbers in $[0, 1]$ contains them all.
- The set of real numbers, $\reals$
-
The set of all subsets of $\nats$
By Cantor’s theorem, no set is in bijection with the set of all of its subsets.
-
The Cantor set
Even though its total length is $0$, its points correspond to the infinite sequences of $0$s and $2$s in base $3$, which are uncountable.
Non-examples
-
The empty set $\emptyset$
(definition)
The empty set is not uncountable, because it is finite.
- The integers $\mathbb{Z}$
- $\nats$
- The set of rational numbers, $\mathbb{Q}$
-
$\mathbb{Q} \cap [0, 1]$
$\mathbb{Q} \cap [0, 1]$ is bounded, because all of its elements are between $0$ and $1$, and it is not uncountable.
-
$\setdef{1, 2, 3}$
$\setdef{1, 2, 3}$ is not uncountable, because it is finite.
Properties
- $\reals$ is uncountable (Cantor’s diagonal argument), and so is every interval with more than one point.
Examples
| Example | infinity | infinite | bounded | divergent | countably_infinite | uncountable | Why |
|---|---|---|---|---|---|---|---|
| $\nats$ | – | ✓ | ✗ | – | ✓ | ✗ | |
| $\mathbb{Q} \cap [0, 1]$ | – | ✓ | ✓ | – | ✓ | ✗ | |
| $[0, 1]$ | – | ✓ | ✓ | – | ✗ | ✓ | |
Missing example: sequence_alternating | |||||||
| The sequence $1, 2, 3, \dots$ | – | – | ✗ | ✓ | – | – | |
Missing example: infinity | |||||||
Notes
- A set is unbounded if it does not fit in any ball: for every $M > 0$, it has two points farther than $M$ apart. This requires a notion of distance, whereas being infinite does not.
- “Infinitely many” and “arbitrarily large” are different. The set $\setdef{1/n \suchthat n \in \nats}$ has infinitely many elements, none larger than $1$.