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In the Riemann sphere model, people extend the complex plane with a point at infinity: $\mathbb {C} \cup \{\infty \}$ and claim this expression: $\frac{1}{0}=\infty$.

I'm confused with what is this $\infty$. Is it a member of $\{\aleph_n\}$ numbers? When we study infinities, we usually discuss cardinality. Does this $\infty$ corresponds to something's cardinality?

Xander Henderson
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LePtC
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    In response to the edit, I don't think I would agree that "we usually discuss cardinality" when we discuss infinity. People often discuss things like ${\displaystyle \lim_{n\to\infty}}a_n$, too. For a list of at least ten different concepts associated with infinity (of which cardinality is just one), see my answer to the MathSE question Understanding Infinity. – Mark S. Mar 07 '21 at 16:31

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Does it has a cardinality?

No. The $\infty$ there doesn't represent a particular set (though you might have to encode it as a set if you're using set theory as your foundation), nor the cardinality of a particular set.

What's the property of this infinity?

The properties are discussed throughout the English Wikipedia article for "Riemann sphere". Is there a particular property you have a question about?

Mark S.
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