TITLE: Types of infinity
QUESTION [21 upvotes]: I understand that there are different types of infinity: one can (even intuitively) understand that the infinity of the reals is different from the infinity of the natural numbers. Or that the infinity of the even numbers is the same as that of the natural numbers. How many types of infinity are there? Or is this infinite itself?

REPLY [2 votes]: I should point out that your examples are all the same type of infinite number: they are all cardinal numbers, measuring an aspect of sets that extends the notion of "how many?" There is a related notion of ordinal numbers which extend the notion of counting.
But there are other types of infinite numbers. One family of related concepts includes the extended real numbers $+\infty$ and $-\infty$ that you might see in calculus. These two infinite numbers quantify something very different than what cardinal numbers quantify. The geometric idea they express is that $+\infty$ marks one "end" of the real number line, and $-\infty$ marks the other "end". There are a variety of other examples in this family, such as the projective real line, and the geometric concept of points at infinity (e.g. see the projective plane or read about elliptic curves).
Another type of infinite number is probably better expressed in relation to infinitesimal numbers: objects that behave as if they were smaller in magnitude than any non-zero real number, but yet were non-zero themselves. In some such number systems (e.g. the hyperreal numbers), one can take the reciprocal of a non-zero infinitesimal number, and the result is an infinite number.
All three of these types of infinite numbers behave very differently and are used for very different purposes.