Hilbert's paradox of the Grand Hotel

In mathematics, the German mathematician David Hilbert (1862 – 1943) presented the following paradox about infinity:

Application to the Cosmological Argument for the existence of God

A number of defenders of the cosmological argument, among others William Lane Craig, for the existence of God have attempted to use Hilbert's hotel as an argument for the physical impossibility of the existence of an actual infinity. Their argument is that, although there is nothing mathematically impossible about the existence of the hotel (or any other infinite object), intuitively (they claim) we know that no such hotel could ever actually exist in reality, and that this intuition is a specific case of the broader intuition that no actual infinite could exist. They argue that a temporal sequence receding infinitely into the past would constitute such an actual infinite.

Related Topics:
Cosmological argument - William Lane Craig

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

However, the paradox of Hilbert's hotel involves not just an actual infinite, but also supertasks; it is unclear whether this claimed intuition is really the physical impossibility of an actual infinite, or merely the physical impossibility of a supertask. A causal chain receding infinitely into the past need not involve any supertasks. See Thomas Aquinas' Summa Theologica for details about infinite regressions and the existence of God.

Related Topics:
Thomas Aquinas - Summa Theologica

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It should also be noted that the addition of guests to a full Hilbert's would require infinitely fast communication, in order for every guest to tell the next guest to move one up in a finite amount of time. Thus, a universe could contain an actual infinite hotel, but with a finite speed of light, and hence it would not be able to contain any more guests even if it was full. (Subtraction, however, does not suffer from this supposed "dilemma." Suppose that each odd guest spontaneously decided to step outside of his or her room at the same moment. Thus, half of the rooms would be cleared instantaneously. In fact, addition can be treated in the same way. Suppose that all of the even guests happened to be standing right outside of the doors to their rooms, and they all, independently of each other, entered at the same time. Thus, half of the rooms could be filled instantaneously. Thus, there is a dilemma only if infinitely fast communication is required in order for the guests to perform these actions, which it is not.)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This is debatable — if the infinite hotel was a single row of rooms, connected by a long hall, each new guest could be placed in the room nearest the front desk, and told to instruct the inhabitant of room 0 to move to room 1, and pass along a similar message, in a way similar to the working of an infinite systolic array. Then, more guests could always be added — but at a rate limited by the walking speed of the guests, their ability to gather their possessions, etc. However, in that case, the hotel would always be overfull (in an overfill, the number of rooms is exhausted, but guests have been forced to share rooms, or some are deprived of rooms). And of course, you can add more guests to a full finite hotel as well, making it overfull.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~