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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$\begingroup$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
$\endgroup$ $\begingroup$$\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.
$\endgroup$ $\begingroup$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.
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