Infinite set
Elementary
A list of items is infinite if it never runs out: no matter how many elements you list, there are more.
Undergraduate
Consider a set $S$. The set $S$ is infinite if the number of elements in $S$ is not any natural number.
Graduate (set theory)
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
Examples
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$[0, 1]$
High school
$[0, 1]$ is uncountable, and so infinite, because Cantor’s diagonal argument shows it cannot be listed.
-
$\nats$
High school
$n \mapsto 2n$ is a bijection from $\nats$ onto the even numbers, a proper subset.
- $\mathbb{Q} \cap [0, 1]$ High school
- The set of all subsets of $\nats$ Undergraduate
Distinctions involving this term
- 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.