A nice illustration of some of the simpler properties of
(countably)
infinite sets.
An infinite hotel with rooms numbered
can be full and yet have
a room for an additional guest. Indeed, simply shift the existing guest in
room
to room
,
the one in room
to room
,
etc. (in general, the one in room
to room
),
to free room
for the newcomer.
There is also room for an infinity of new guests. Indeed, shift the
existing guest in
room
to room
,
the one in room
to room
,
etc. (in general, the one in room
to room
),
to free all rooms with odd numbers for the newcomers.
These examples illustrate that an infinite set can be in bijective
correspondence with a proper subset of itself. This property is sometimes
taken as a definition of
infinity
(the
Dedekind definition of infinity;
see also
Infinity).