bellhop or to infinity and beyond

Gizmodo offers a challenging but rather intriguing primer on the nature of infinity—which is not a number itself or some threshold, unless posited as the point at which parallel lines verge together, and the idea that infinity is amenable to being doubled or tripled through a quantum mechanical demonstration that makes a classic thought experiment seem not so rarefied or cheeky.  In 1925, mathematician David Hilbert pondered the following brain-teaser: supposing there is a grand hotel with an infinite number of rooms which is always booked and has no vacancies, but a guest desperately begs in the lobby for a room for the night. The hotel staff can still oblige, despite the occupancy and the infinite inconvenience, since in a countable infinity, there is always a +1, by have the guest in room number one move into room number two, and so on. By a countable infinity, and there are several different types of infinities, Hilbert means an enumerated set, that one could walk the corridors counting off room after room—though one might never reach the end—and also room-service is not logically bamboozled as they know the new whereabouts of every visitor and N+1. Then suppose an infinitely large tour bus with an infinite number of guests pulls up in the parking lot. No problem still, says management, as everyone in an odd numbered could move to an even numbered one and the vacated rooms—bogglingly, free up accommodations for the infinite number of new arrivals. This shifting works, logically and in quantum states where vacancies are created, because the countable infinity—once taking on more guests, while still assigned to a numbered room, Hilbert’s Hotel becomes another sort of infinity—the kind that is innumerable, something that can’t be counted in a discrete way because there always room in between—like the number of points on a line—being infinite and a point being that which has no part, something dimensionless. Paradoxical things may appear only academic when first puzzled to their conclusions but it is pretty astounding and reassuring to find that there is potentially real application for these concepts.