Infinite

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A set $S$ is said to be infinite if there is a surjection $f:S\to\mathbb{Z}$. If this is not the case, $S$ is said to be finite.

In simplified language, if a set is infinite, that means that it doesn't end, i.e. you can always find another element that you haven't examined yet.

Equivalent formulations

  • A set is infinite if it can be put into bijection with one of its proper subsets.
  • A set is infinite if it is not empty and cannot be put into bijection with any set of the form $\{1, 2, \ldots, n\}$ for a positive integer $n$.
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