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A set $S$ is infinite if it is not finite; that is, there is no $n\in\mathbb{N}$ for which there is a bijection between $n$ and $S$.
Assuming the Axiom of Choice (or the Axiom of Countable Choice), this definition of infinite sets is equivalent to that of Dedekindinfinite sets.
Some examples of finite sets:

The empty set: $\{\}$.

$\{0,1\}$

$\{1,2,3,4,5\}$

$\{1,1.5,e,\pi\}$
Some examples of infinite sets:

$\{1,2,3,4,\ldots\}$.

The primes: $\{2,3,5,7,11,\ldots\}$.

The rational numbers: $\mathbb{Q}$.
The first three examples are countable, but the last is uncountable.
Keywords:
infinite
Related:
Finite, AlephNumbers
Synonym:
infinite set, infinite subset
Type of Math Object:
Definition
Major Section:
Reference
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9 is not prime by mathwizard ✓
Linking by mathwizard ✘
synonym by matte ✓
wrong statement by scineram ✓
(countable) axiom of choice by yark ✓
Linking by mathwizard ✘
synonym by matte ✓
wrong statement by scineram ✓
(countable) axiom of choice by yark ✓