On Sun, 23 Feb 2014 17:21:35 -0000, "Julio Di Egidio" <firstname.lastname@example.org> wrote:
>"John Gabriel" <email@example.com> wrote in message >news:firstname.lastname@example.org... >> On Sunday, 23 February 2014 08:23:54 UTC+2, Virgil wrote: >>> In article <email@example.com>, >>> "Julio Di Egidio" <firstname.lastname@example.org> wrote: >>> >>> > "Consider a hypothetical hotel with a countably infinite number of >>> > rooms, >>> > all of which are occupied." (*) >>> >>> > The nature of countable infinity is such that there cannot be a last >>> > room, >>> > and that's fundamental to the fact that new guests can be accommodated >>> > in an >>> > infinite hotel at will, itself an illustration of the fact that we can >>> > count >>> > endlessly. But, if there is no last room, it can never be the case >>> > that >>> > *all* rooms are occupied, hence the whole argument, for how informal, >>> > falls >>> > apart since inception. >>> >>> One can have a bijection between two infinite sets, such as between the >>> odd natuals and the even naturals. >>> >>> A bijection between rooms and occupants would leave no room unoccupied >>> and no occupant unroomed.. >> >> Hogwash. A bijection does not lead to that conclusion at all. > >Agreed. In fact, "no room" is the converse of "all rooms", and Virgil is >simply missing the point. > >> A bijection exists iff a transformation is one-to-one and onto. >> This means that f is a bijective function if f:x->y <=> finverse (y) = x >> for all x in the domain of f. >> >> The catch: There is no such thing as f(oo) = k and finverse(k) = oo. > >Indeed. To reiterate, mine was not an objection to the notion of 1-to-1 >mappings, e.g. I have no qualms with the idea that there are as many even >natural numbers as there are natural numbers: both collections are simply >*endless*. It is the idea that the natural numbers form a set, i.e. a >complete totality that is, as I am contending, untenable.
Why? I don't see the contradiction between "endless" and "set". Am I missing something?
> >The consequence, again, is that Hilbert's hotel is illogical and, in fact, >misleading: either there is always room left, or all rooms are occupied, but >not both. IOW, if an infinity of rooms is occupied by an infinity of >guests, there is no room left, and the trick of moving each guest to the >next room cannot be performed: 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.
Y'know that *sounds* convincing, logically, in English and probably in many other natural, human languages, but a lot of things that *sound* convincing just are not. Hilbert's flying saucers are one. Infinities are fun to play with (unless your "work" is firmly based on (3 * (10 ^ 603)) being "infinity" in which case they are just bizarre) but they pretty much don't fit standard languages.
"In back, cats have two legs, and in front they have forelegs. That's six legs, which is an odd number of legs for a cat. However, it's well known that cats have an even number of legs. No finite number is both even and odd, so every cat has an infinite number of legs.".
Okay, so the joke is easily seen through when typed in English but the humour of that little nonsense paragraph can be replicated all through mathematics where infinity touches it.
I really don't think we should be thinking of infinity in terms of Hilbert's Hotel, flying saucers or anything else in standard languages. That just confuses the picture, though it is great fun to do this. Maybe we should just stick to using the mathematics? J.