h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I can make the Infinite Hotel turn away customers. I'll simply come there with a tour group which has one member for every irrational number (such as "pi").

The reason this works is... there are ascending orders for infinity. The Infinite Hotel falls into the order which contains only rational numbers. The order which contains irrational numbers is higher. In other words, you can at least *begin* to count the rational numbers, but you can't even do that for the irrationals.

I'm sure once I open the Irrational Infinite Hotel next door, someone can swamp that too, but I don't know how.

To swamp the irrationally infinite hotel (with aleph-1 rooms) you need a busload of people where there is one person for each possible set of irrationals. IE, one for {1,2,3}, one for {}, one for {pi, e, root 2}, etc, etc. (aleph-2 people).

The profit for infinite hotels is undefined - so they make life very hard for the taxman. That's why you don't see so many of them.

There is one slight problem in the idea of an Aleph-1 hotel prevents it from existing. Okay, you could say that Hilbert's hotel can't exist either, as it is infinitely large, but the Aleph-1 hotel can't even exist in the way that Hilbert's does exist.
Why? As everyone following these threads must be aware of by now, all infinities EXCEPT for Aleph-0 are uncountable. This means that you can't enumerate the elements. Thus, it is impossible to have a hotel with Aleph-1 rooms, as there is no way you could even start placing the rooms one after another. In the same way, Aleph-1 guests could never arrive.
Unless (and this is a quite interesting unless) there is a _Continuum_ of rooms and/or guests. What this would mean in practice (heh, silly word in in this kind of conversation) is that all rooms (or guests) sort of "flow" into each other - there is no boundary between one room/guest and the next. It makes things awfully tricky, ugly and just generally confusing though...plus it all gives me a feeling of discussing some very awkward form of pantheism...

"There's no privacy in an aleph-1 hotel"

I love pure maths for it's ability to generate nonsense...

However, although you can't do it in this universe (or any universe with a countable number of dimensions, you can make a hotel with an uncountably infinite number of rooms in a space with an uncountable number of dimensions. Easy peasy!