Hilbert's Hotel

We all think we know what it means when we talk about something continuing "into infinity". However, the philosopher  William Lane Craig identifies two kinds of infinites:

Potential infinite
A potential infinite is when we can keep adding items to something without stopping.

Actual infinite
An actual infinite is when the number of items has reached its total infinite number. Craig thinks this kind of infinite is impossible. He suggests that the 'Hilbert's Hotel' thought experiment shows that actual infinites are absurd.

We are used to using the idea of infinity as a useful concept in mathematics, but which kind of infinity are we talking about in the  Cosmological Argument?

What does this have to do with the Cosmological Argument?
Premise option 2a) of the argument offers the choice of an infinite regression of causes. If it were possible, such a regression would be an actual infinite.
The absurdity of Hilbert's Hotel seems to show that actual infinites are impossible.
Therefore, in the cosmological argument, we are left with Premise option 2b): that there is some uncaused thing which caused everything else.

This thought experiment was devised by the German mathematician David Hilbert (1862-1943).
Imagine a hotel with an infinite number of rooms. It's a busy bank holiday weekend and all the rooms are full.

A new guest arrives and wants to check in. The clerk says, "No problem!" and moves the guest in room 1 into room 2, the guest in room 2 into 3, and so on. The hotel is no longer full (even though it still has an infinite number of guests), and the new guest moves into the empty room 1.

Although the hotel has an extra guest on the register, the total number of guests on the register remains the same, and the hotel is full again.

However many new guests turn up in the lobby, the hotel will always be able to accommodate them.

Even if an infinite number of guests turn up wanting rooms there's still no problem: the clerk simply moves the guests in odd numbered rooms up (1 to 3, 3 to 5 etc.) to create an infinite number of empty rooms, and the new guests move in.

What happens to the total number of guests if all the guests occupying even numbered rooms move out? The hotel is then half empty, but still has an infinite number of guests.

And finally: how long would new guests have to wait for the clerk to make the registrations and room changes?

Think all this is impossible? Craig agrees with you. This is because when the thought experiment talks about the hotel being full and the total number of rooms and guests being infinite, we're dealing with actual infinites, which Craig says are impossible.

© Mr.B at Woodford County High School