"Julio Di Egidio" <firstname.lastname@example.org> wrote in news:email@example.com:
> "Wizard-Of-Oz" <firstname.lastname@example.org> wrote in message > news:XnsA2DE594FBC0E6somewhereovertherain@18.104.22.168... >> "Julio Di Egidio" <email@example.com> wrote in >> news:firstname.lastname@example.org: >>> "Julio Di Egidio" <email@example.com> wrote in message >>> news:firstname.lastname@example.org... >>>> >>>> slightly more rigorously, if we have 1, 2, 3, etc. up to >>>> (allegedly) *all* natural numbers, there is just no natural number >>>> missing that we can add to the lot. >>> >>> If, for all n, room n+1 is occupied already, the hotel is just full >> >> Badly expressed, as your condition fails if the first room is >> unoccupied. > > I had assumed some context, pardon my naiveté: but even conceding the > nitpick, the argument does not fail, one single room available is just > not enough to salvage our hotel.
It requires no salvaging
>> You should have said that for all n, room n is occupied. > > No, it's rather fundamental to the implied inductive argument that one > talks of a situation at n+1 as a consequence of a situation at n... > Never mind, you have a point: I will try and rewrite it more > rigorously.
>>> and no more guests can be accommodated. >> >> That's where you're wrong when the hotel has infinite rooms > <snip> > > That's rather where you start paraphrasing the usual thesis with no > argument at all.
It is very simple to map guests to rooms. There are many ways of doing this. One at least results in there being no room left unoccupied. But not all mapping require that every room has a guest.
So simply changing the mapping (ie reassigning which guest is given which room) can allow for as many vacancies as you like .. even an infinite number of them
Is there something about the above with which you disagree?