Suppose
that the hotel is, indeed, full – there are people in every room – that that
one new guest arrives. There is surely no problem involved in placing the new
guest in room 1, moving the guest in room 1 to room 2, moving the guest in room
2 to room 3, and so on. But, plainly enough, other guests will die (or move
out) long before they are asked to change rooms. Once this is seen, we can note
that – for this particular problem, namely, accommodating a new guest in a
hotel that has no empty rooms – the very same strategy could be used if the
hotel were finite but extremely large. Of course, those guests who are asked to
change rooms in the middle on the night will not be happy – but this doesn’t
count at all against the feasibility of making accommodation time for each of a
large number of already accommodated guests.
The
general point to be made here is that mere acceptance of the possibility of a hotel
with infinitely many rooms does not commit one to acceptance of the possibility
of manipulating all of the infinitely many rooms in a finite amount of time.
For all that has been argued so far, it might be that one can accept that there
can be a hotel with infinitely many rooms while also denying that one can
accommodate a new guest by moving the occupants of room N to room N +1
(for all N).
Suppose,
for example, that we live on a plain that extends to infinity in all
directions. Suppose, further, that there is a building, with rectangular
cross-section, which has a façade at a given location, and then extends to
infinity from that façade. The rooms in the hotel may be supposed to be
assembled into groups of one hundred, each governed by its own sub-reception.
The sub-receptions, in turn, may be supposed to belong to groups of one
hundred, each governed by an administrative office (the first of which will
thus be about 10,000 rooms from the façade). These administrative offices, in turn,
may be supposed to belong to groups of one hundred, and so on (2). If a guest
turns up to the sub-reception nearest to the façade at a time at which there
are no unoccupied rooms in the hotel, then it may well be possible to
accommodate this guest by slightly inconveniencing many other guests – but there
is nothing at all in this story that forces us to allow that the guest can be
accommodated by moving the occupants in room N to room N + 1 (for
all N).
Suppose
we grant that the hotel might be constructed in such a way as to prohibit the
various manoeuvres that are taken to be problematic by some foes of large,
denumerable, physical infinites. Plainly, it does not follow that the hotel
cannot be constructed in such a way as to permit these various manoeuvres. For
instance, we might suppose that the hotel is constructed according to the
following plan. The hotel is a skyscraper, in which each floor is half the size
of the previous floor. Moreover, inside the hotel, things halve in size as they
move up from one floor to the next; and the elevator – from the standpoint of
those inside the hotel – doubles its average speed as it passes from one floor
to the next, as do all other things that are in motion. Given these amendments –
or, at any rate, given these kinds of amendments – such things as the
checking out of infinitely many guests can be accomplished in finite time. Of
course, there are various ways in which this kind of story contradicts known
physics – for example, it supposes that there is no smallest quantum of energy
or matters, that there is no upper limit to particle velocities, that matter is
stable under arbitrarily large accelerations and decelerations, and so forth –
but it should not be supposed that this is a reason for denying that the
envisaged scenario is metaphysically possible.
I have
not tried very hard to give a detailed account of a hotel that will permit the
checking out of infinitely many guests in a finite account of time. While I am
fairly confident that this could be done, it is not necessary for present
purposes. For the foes of large, denumerable, physical infinites clearly face a
dilemma at this point. On the one hand, if there is a way of carrying out such
a detailed account, then the friend of large, denumerable, physical infinites
win by outsmarting his foes: There can, after all, be a hotel in which
infinitely many new guests are accommodated, even though all rooms are full,
via the simple expedient of moving the guests in room N to room 2N
(for all N). On the other hand, if there is no way of carrying out such
a detailed account, then the friend of large, denumerable, physical infinities
wins by claiming that, despite the fact that there can be a hotel with infinitely
many rooms, the various manoeuvres that the foe of large, denumerable, physical
infinites takes to be impossible are, indeed, impossible. (Graham Oppy, Philosophical
Perspectives on Infinity [Cambridge: Cambridge university Press, 2006], 51-53)
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