Math Investigation eTwinning. Mission: Infinity...Affairs! Volume: Infinity | Page 18

Fill in the gaps with the words listed below: (pay attention: not all the words are necessary!)
cardinality - integers – passengers - hotel – Set Theory – seats - kind - real numbers – math empty - guests – paired up - countable – size - Continuum Hypothesis – full - numbers –
mathematician – room –move –cardinality - infinity
Infinity is without doubt one of the hardest mathematical concepts to understand.
The mathematician David Hilbert tried to explain some of the properties of _______ using a hotel
with infinitely many rooms. We could imagine that it consists of just a single, never-ending
corridor, with the rooms numbered 1, 2, 3, 4 and so on. On the day we arrive at this _____, every
single room is full. In any other hotel, the reception would have to turn us away.
Here, instead, once we ask for a _____, the hotel asks all the guests to move up into the next room.
Since there is no last room in the hotel, every guest will get a new room.
Now the first room is _____, and we can move in. The Hilbert doesn’t have a last room!
This idea can be extended: thus even if the hotel is ____ it can still offer rooms to any finite number
of new guests!
Let us suppose that the hotel is full again, and that infinitely many new guests arrive. Now the
previous method won’t work anymore: you can’t ask the ______ to move up infinitely many rooms.
Instead the hotel asks the guests to move into the room with twice the number of their current room.
Finally let us suppose that a whole bus company arrives at the hotel: they have infinitely many
buses, each with infinitely many _____. Even if it’s more difficult, we can also move all those
infinitely times infinitely many passengers into this already full hotel.
If we want to go even further, we have to develop a new language that is called ___________.
A set is simply a collection of any objects. We say that two sets have the same size if we can build a
bijection pairing up their elements so that no element is left out or connected to two elements. This
applies