Infinite

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, a set is infinite if 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 $n$.

Applications to Infinity with Sums

A sum works the same way. Certain sums equate to infinity, such as

$\sum_{i = 3}^{\infty}{(2i - 1)}$

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"Operations" with Infinity

Some bad rules involving operations with infinity are as follows:

  • $1/{\infty} = 0$
  • ${\infty} + x  = {\pm}{\infty}$
  • ${\infty}\cdot{x} = {\infty}$

None of these are true because $\infty$ is not a real number which you can write equations involving.

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