When we talk about infinity, we never really talk about it as though it were an actual value. In calculus, we really only talk about limits approaching infinit

When we talk about infinity, we never really talk about it as though it were an actual value. In calculus, we really only talk about limits approaching infinity (sometimes using "infinity" as a shorthand). In set theory, we really only talk about things having infinite cardinality (sometimes using "infinite" as a shorthand). In geometry, we talk about things being "infinite" not to suggest that there's some endpoint that is simply unreachable; we mean something more along the lines of "unbounded." Hilbert's hotel is more of a set-theory problem than a calculus or geometry problem, so it's best to think of the set of rooms having infinite cardinality. A set of countably infinite cardinality (the guests) is of the same cardinality when another (finite or) countably infinte set is added, so we can develop a one-to-one correspondence between it and another set of countably infinite cardinality (the rooms). Thinking about it this way demystifies the problem a little.