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Topic: The self-contradictory infinite hotel
Replies: 62   Last Post: Feb 24, 2014 10:17 PM

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 LudovicoVan Posts: 4,165 From: London Registered: 2/8/08
Re: The self-contradictory infinite hotel
Posted: Feb 23, 2014 12:21 PM

"John Gabriel" <thenewcalculus@gmail.com> wrote in message
> On Sunday, 23 February 2014 08:23:54 UTC+2, Virgil wrote:
>> In article <lebotv\$fc7\$1@dont-email.me>,
>> "Julio Di Egidio" <julio@diegidio.name> 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.

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.

Julio