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Can infinity be an element of a set? For example:

{$1,2, \infty,4$}

I understand $\infty$ is not a real number

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    "A set" can contain anything. {1,2,Elephant,4} is a set. – billpg Feb 09 '15 at 14:45
  • @billpg: it's not quite as simple: you do need to properly define what you mean by the things you list as set elements. A slight distortion of your example shows this very clearly: if ${1,2,\text{Elephant},4}$ is a set, what is ${1,2,\text{Elephant},4,\text{Elephant}}$? It would certainly also be a set, but it's totally not clear whether it would contain two different elephants, or rather the value $\text{Elephant}$, whose redundant appeareance in the set-builder doesn't make any difference. Similarly, ${1,\infty}$ doesn't make sense if $\infty$ was never defined as such. – leftaroundabout Feb 10 '15 at 00:33

3 Answers3

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If you're talking about as a subset of real numbers, no. If you're talking about an arbitrary set, then sure, as long as you define what you mean by $\infty$. For instance, you could mean it as the one point added in the one point compactification of the real numbers, then it certainly can be an element of a set

Alan
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$\infty $ is indeed not a real number. However, sets don't have to only have real numbers as elements. So, if you want, you can add the symbol $\infty $ to any set you like. You can define the meaning of $\infty $ in various ways too.

Ittay Weiss
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As a relevant side remark: An interesting treatment of $\infty$ is the one point compactification, for example the Riemann sphere $\mathbb{P}^1\sim \mathbb{C}\cup\{\infty \}$. This could also be done on $\mathbb{R}$ by considering $S^1$ projecting onto $\mathbb{R}$. You can google one point compactification for details.

Qidi
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