They assume that the absurd does not have the ability to be. they assume the hotel is absurs

A great question as far as I'm concerned. Reminds me of @Metaphysician Undercover's thread on coherentism. Surely, if every room in Hilbert's hotel is occupied, it's impossible to accommodate more guests. Yet, Hilbert's hotel does have room for not only a finite number of guests but also an infinite number of them. A seemingly unresolvable paradox ensues: there's no room for more guests and there's room for more guests.

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.

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? — jay232

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?

Let's face it: Despite their seductive allure, we have no direct observational evidence for either the infinitely big or the infinitely small.
...
Not only do we lack evidence for the infinite but we don't need the infinite to do physics. — Edge - 2014: What Scientific idea Is Ready For Retirement? - Infinity - Max Tegmark

If there is an infinite number of guests at the hotel already, then there are no more possible guests to arrive, because that would allow for the possibility of greater than infinite.

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? — Andrew M

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?!"

The Hotel doesn't derive a contradiction, it's not technically a paradox as such.
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.

Bears noticing: ∞ ∉ R — jorndoe

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?
— Andrew M

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. — SophistiCat

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".

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? — Andrew M

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.

Whoever Craig and Mooreland are, they might wish to take a college-level course in mathematics. There’s no paradox that I can see here, only a metaphor for some bijections from N to a subset of N.

Whoever Craig and Mooreland are... — Olivier5

They are perhaps a pair of fundamentalist christian theologians. 'nough said.

If there is an infinite number of guests at the hotel already, then there are no more possible guests to arrive, because that would allow for the possibility of greater than infinite. — Metaphysician Undercover

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.

They are perhaps a pair of fundamentalist christian theologians. 'nough said. — Banno

Mmmmmokay..... <:mask:

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. — TheMadFool

The power set of the naturals.

The power set of the naturals. — jgill

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.

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. — jgill

I don't know what a power set is and I don't see anything "unusual" like that going on in Hilbert's hotel

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?
— Andrew M

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. — SophistiCat

:up:

Whoever Craig and Mooreland are, they might wish to take a college-level course in mathematics. There’s no paradox that I can see here, only a metaphor for some bijections from N to a subset of N. — Olivier5

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:

Hilbert's Hotel is absurd. Mind you, it's logically correct for the mathematician but it's impossible for something like Hilbert's Hotel to really exist. You can describe it on paper but it cannot exist in reality. Illustrations like these showed that the existence of an actually infinite number of things is impossible.

Now sometimes people react to Hilbert's Hotel by saying that these paradoxes result because we can't understand the infinite - that it's just beyond us. But this reaction is in fact mistaken and naive. Infinite set theory is a highly developed and well understood branch of modern mathematics.

These absurdities result not because we do not understand the infinite but because we do understand the nature of the actual infinite. Hilbert was a smart guy and he knew well how to illustrate the bizarre consequences of an actually infinite number of things. — Hilbert's Hotel and Infinity - William Lane Craig (from 4:30)

Hilbert:

We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. Can thought about things be so much different from things? Can thinking processes be so unlike the actual processes of things? In short, can thought be so far removed from reality? Rather is it not clear that, when we think that we have encountered the infinite in some real sense, we have merely been seduced into thinking so by the fact that we often encounter extremely large and extremely small dimensions in reality?
...
In summary, let us return to our main theme and draw some conclusions from all our thinking about the infinite. Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought — a remarkable harmony between being and thought. In contrast to the earlier efforts of Frege and Dedekind, we are convinced that certain intuitive concepts and insights are necessary conditions of scientific knowledge, and logic alone is not sufficient. Operating with the infinite can be made certain only by the finitary.

The role that remains for the infinite to play is solely that of an idea — if one means by an idea, in Kant's terminology, a concept of reason which transcends all experience and which completes the concrete as a totality — that of an idea which we may unhesitatingly trust within the framework erected by our theory. — On the infinite - David Hilbert

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. — TheMadFool

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.

I dismiss Cantor as misunderstanding. — Metaphysician Undercover

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.

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. — Andrew M

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.

I've got one in my backyard. It's infinitely small — Metaphysician Undercover

Does it help reduce cleaning cost?