It's worse than that. "Infinity" isn't even a single concept. There are at least two distinguishable uses of infinity (there may be more, but I haven't figured

It's worse than that. "Infinity" isn't even a single concept. There are at least two distinguishable uses of infinity (there may be more, but I haven't figured them out yet, not that my opinion counts for much). There's the adjective "infinite" that refers to a property of sets. This is the type that Cantor studied, and it turns out to have many different types, which are pretty strictly ordered into layers. Then there's the noun "infinity" which is either a point or a location that points can exist at, and while it's usually possible for there to be several such infinities in several different directions, they don't come in layers. I believe Rider of Giraffes refers to these as "set-theoretic" and "geometric" infinities, respectively. For the first definition, it's easy to distinguish between it and traditional numbers: an infinite set is bijective to a proper subset of itself. However, it's also useful to consider it a generalization of numbers, so you can count forever. The second definition is more problematic. Here, infinity is just a point, just like all your other points. You can choose to add it to your set, or not. It usually behaves a little funny (like it makes certain operators not invertible), but you may have to get subtle to define it, or a set that contains it. Sometimes, it's not different than normal points at all, and sometimes it depends on the context. For example, if you take the real line and add +/-infinity, in topology you just get a closed interval like [0,1], whereas in analysis based on metric spaces you get something outright broken (fails to satisfy the axioms of the objects being studied).