David Hilbert’s thought experiment known as ‘Hilbert’s Hotel
Why Craig and Mooreland think that the existence of Hilbert’s Hotel would be absurd?
Why they think the absurdity of Hilbert’s Hotel implies that no actual infinite collection can exist?
Why they think the absurdity of Hilbert’s Hotel implies that no actual infinite collection can exist?
Comments (25)
Here's how I look at it, if it's any help. First thing to remember is that Hilbert's Hotel has infinite rooms and that they're all occupied by an infinite number of guests.
Then imagine X number of guests arrive, looking for rooms in Hilbert's hotel. Observe:
X can range from 1 to infinity. The basic equation for the total number of guests, T = X + infinity (the infinite number of guests already occupying the hotel)
No matter what the value of X, T + X = (still) infinity. In other words, the total number of guests, those who are already in the rooms and those who want rooms) doesn't change - it's always infinity and how many rooms does Hilbert's hotel have? Infinity, exactly the number of rooms we need.
I don't know why they think that. But if it's to be a thought experiment about the physical world, then we have no experimental evidence that there is, or can be, anything infinite. And what would such an experiment look like? How would it be measured?
Quoting Edge - 2014: What Scientific idea Is Ready For Retirement? - Infinity - Max Tegmark
Same way as how we establish anything in science: that Earth is ~4.5 Gyr old ("How could you possibly know? Were you there?!"), that pulsars are neutron stars, etc. We develop models and evaluate their closeness of fit, simplicity, and other epistemic and scientific virtues.
In the Edge essay Tegmark faults infinities for some specific outstanding problems in cosmology, such as the measure problem, and suggests that alternative finitistic models would do better. (Needless to say, this is not as simple and uncontroversial as it sounds.) Notice how this argument fits with the general theory selection process that I outlined above, as opposed to "Gosh! Infinities are so paradoxical!" or "Gosh! How could we ever measure infinity?!"
But it does derive counter-intuitive implications.
Oddly enough perhaps, some finites also derive counter-intuitive implications.
WL Craig and JP Moreland use terms like absurd about an infinite temporal past in particular.
(I'm guessing intuition is sometimes more or less the principle of sufficient reason.)
Bears noticing: ? ? R
No, infinites aren't just more numbers (apparently this keeps escaping many folk); you don't add subtract compare them all and call it a day; some rigor is required here, and we already know this.
Which is to say that principles other than experiment and measurement are appealed to.
We can conduct experiments to determine a specific finite age of the Earth. But how would we test whether something was infinite in age, size or number as opposed to just really, really large?
We already know that things with finite ages, sizes and number exist in nature. But, as Tegmark points out, "The assumption that something truly infinite exists in nature ... [is] an untested assumption".
Obviously, not by counting or measuring directly. We don't hold a stopwatch to measure the age of the earth either - we use other measurements to establish theories in which the age of the earth is a bound variable. Same with the size of the universe: it makes a difference to the theories that we use to explain astrophysical observations - their accuracy, simplicity and compatibility with other well-established theories. You can't just arbitrarily choose a size without breaking a bunch of stuff.
They are perhaps a pair of fundamentalist christian theologians. 'nough said.
That's the odd bit. As per Cantor, infinite sets can differ in cardinality i.e. one can be "greater" than the other. The only instance of that I'm familiar with is the set of real numbers, a bigger infinity than the set of natural numbers. Hilbert's hotel, the way it's formulated, seems to restrict itself to the set of natural numbers. In other words, the infinite set that matters in Hilbert's hotel is basically the set of natural numbers and I don't remember ever coming across a claim that there's an operation we can perform on the naturals that can yield a greater infinity than it.
Mmmmmokay..... <:mask:
The power set of the naturals.
Thanks. I didn't word it well. I meant nothing that happens to the naturals in Hilbert's hotel can change the cardinality of the set of natural numbers.
Probably. But take for example the element of the power set of N: {2,6,7}. This could be interpreted as guest(2)->room(6), guest(6)->room(7),guest(7)->room(2). Kind of silly, I guess.
I don't know what a power set is and I don't see anything "unusual" like that going on in Hilbert's hotel
:up:
It seems that Craig is following Hilbert (and others) on this, which is to make a distinction between the mathematical idea of infinity, which he accepts, and its existence in nature, which he rejects.
Craig:
Quoting Hilbert's Hotel and Infinity - William Lane Craig (from 4:30)
Hilbert:
Quoting On the infinite - David Hilbert
I dismiss Cantor as misunderstanding. He misunderstands the nature of numbers, the application of numbers in counting, and the infinite possibility which we say counting gives us. So his theorem is a fine example of category error, in classifying an empty set as a countable set.
Hilbert's Hotel proposes an infinite set, the hotels, and an infinite subset, the guests at the hotel. The guests are not qualified by anything other than "at the hotel". So they cannot exist anywhere other than at the hotel, and they cannot come and go from the hotel as a guest would do at a normal hotel. That would be equivocation, referring to another meaning of "guest at a hotel" which is other than the meaning given to "guest" in Hilbert's Hotel.
That is why the example is so confusing, we tend to think that a guest at Hilbert's hotel would be the same as a guest at another hotel, a guest which could come and go as desired. But the guests at Hilbert's hotel do not have that freedom, because if a guest could come or go, the criteria of the description would not be fulfilled. So talking about a guest arriving at the hotel is nonsense. The hotel with infinite rooms and infinite guests is a description of a static scenario. To change that scenario, and talk about guests arriving at the hotel, we'd have to change the description. It simply makes no sense to talk about a hotel with infinite guests, and then suddenly there is more guests.
It would make a better thought experiment to ask what we'd have if one of the infinite guest decided to leave the hotel. How many guests would then be at the hotel? This would reveal that there is an inconsistency between finite numbers and proposed infinite numbers, which cannot be reconciled.
How so? He applies the math used by, we could even say, prehistoric hunter-gatherers, [blone-to-one correspondence[/b], which, by the way, is also the foundation of modern number theory.
We've been through this before on other threads, but we could start with the difference between representing an infinite sequence, and representing each of the individuals within an infinite sequence. The first is possible to do, the second is not. Accordingly, we can represent the natural numbers as infinite, but we cannot represent the set of natural numbers because the natural numbers are infinite. Of course we've already seen that this is denied on this forum by the proponents of set theory
Apply this to Hilbert's Hotel. We can represent an infinity of rooms at the hotel, or an infinity of guests, but each room or each guest has its own identity as distinct from each other, to be a separate individual. So we cannot represent each individual room, or each individual guest, in the scenario because there is an infinite number of them, and that would be impossible. therefore we cannot represent the set of rooms at the hotel, or guests at the hotel.
The category mistake I referred to, involves conflating the description of the objects in the set "the natural numbers", and the actual members of the set. Allowing that a set is defined by it's descriptive terms, with defining a set by its actual members, because this produces the empty set, which is incoherent if a set is defined by its actual members A set without members is not a set, if a set is defined by its actual members. But if the set of natural numbers is not defined by its actual members, then it is simply an imaginary set, and not at all related to the real use of numbers.
Well if all they wanted to say is that hotels with an infinite number of rooms cannot exist outside of our imagination, I could have spared them the effort: they don’t exist for obvious reasons of lack of feasibility. Like where would you put them?...
I've got one in my backyard. It's infinitely small though, so the guests have a hard time finding it, and the rooms are even harder to find because they're infinitely smaller, needless to say. So even after the guests find the infinitely small hotel with an infinity of rooms, they have to go through a special procedure to be able to fit into one of those rooms which are infinitely smaller than the infinitely small hotel.
Does it help reduce cleaning cost?