This documentary sounds pretty bad from the description, but I this article paints an incorrect picture of how mathematicans study the infinite. Whether there

This documentary sounds pretty bad from the description, but I this article paints an incorrect picture of how mathematicans study the infinite. Whether there is one infinity or multiple is entirely context dependent. Yes, there's far more than just one infinite cardinal. But it is a mistake to identify "infinities" with "infinite cardinals". In fact, mathematicians use a bunch of systems of infinite numbers; cardinals are only one of them. (Also, they may also use "infinity" in ways that can't reasonably be called numeric, or that are entirely informal; I'm not trying to cover every possibility here, just demonstrate how it's not just cardinals.) For instance, in addition to cardinals, there are also the ordinals, another system of transfinite numbers which works quite differently. Or, if one wants infinitesimals as well, there are the surreal numbers (although these (sort of) include the ordinals). But also, in many contexts, the right thing to do is not to use any of these, and to simply use the extended reals -- i.e., just take the real numbers and just stick a big flat undifferentiated "infinity" on the top of it (and possibly a minus infinity on the bottom depending on context). In the extended nonnegative reals, there really is only one infinity. And as much as I love cardinals and ordinals... they're not appropriate for most situations involving the infinite. The extended reals are useful far more often. You're going to be using a system with only one infinity far more often than a system with multiple infinities. It's worth pointing out the more general point here -- numbers come in systems and cannot be used outside the context of some system! There is not, in fact, any single mathematical definition of a "number". You might imagine that in math there is some set of "the numbers". (Many people seem to specifically have the wrong idea that the complex numbers are "the numbers".) There isn't! There are different systems of numbers and each works differently. Some of these systems of numbers fit together and are compatible with one another; e.g., the whole numbers include into the real numbers. And also, the whole numbers include into the cardinals; it's compatible with either. But some of these systems are not compatible with one another and do not fit together. You can't combine real numbers and cardinals! It's meaningless to ask, "What's 2pi + aleph_0?", even though both are "numbers" in some sense, because these systems are not compatible. (Well, unless you do the work to make them compatible, which can sometimes happen; although note that in many cases systems are quite definitely incompatible and can't be made to fit together.) So it is incorrect to say that cardinals are the way to do infinities and that "infinities" mean "cardinals". There are other, incompatible, systems of infinities; and in many contexts, you want a system with just one infinity, not multiple. And in these contexts, which are numerous, you do just say "infinity" and leave it at that, and this is perfectly correct.