*This is a continuation of "A few things wrong about the cosmological argument," an ongoing series. Today we will discuss William Lane Craig's treatment of Hilbert's Hotel. I will assume you are familiar with Hilbert's Infinite Hotel; if not, you can read William Lane Craig, me, or any other source on the internet.*

William Lane Craig (WLC) thinks Hilbert's Infinite Hotel is absurd. If Hilbert's Hotel is full, then you can add or remove people and it will still have the same number of guests. Absurd!

Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things.This is an argument from absurdity, which has the following form:

A implies B.There is nothing wrong with an argument from absurdity if it is done correctly. But it is not done correctly. I can agree that Hilbert's Hotel is absurd. But not all infinities are necessarily absurd. We can have infinities without specifically having an infinite

Both me and my opponents agree that B is obviously false.

Therefore we should agree that A is false.

*hotel*. Hilbert's Hotel is absurd because we will never have the resources, the manpower to create such a hotel. We don't have the people to fill it, the means to maintain it.

Of course, WLC thinks there is a more abstract property of Hilbert's Hotel which is absurd. Specifically, he claims it is absurd that you can add or remove things, and still have the same number of things. I do not think this is absurd. The argument from absurdity relies on my agreement at this point, but I do not agree.

(An aside: WLC makes a technical error here. We do not say that there is the "same number" of things, we say that the two sets of things have the "same cardinality". Neither set has a well-defined "number" of things, since both sets are infinite. There is a reason we use technical terms like "cardinality", and it is to avoid making mistakes by accidentally applying intuition where it does not apply. WLC may have simply wanted to avoid technical jargon...)

WLC cites a couple people who objected, like me, that infinities are not absurd. His basic response is, "It looks pretty absurd to me." "Nuh uh." "Yeah too." We seem to be at an impasse. If you're keeping track, that means WLC lost, since he is the one presenting the argument, and he has failed to convince. But let's stop keeping track, and focus on resolving the impasse.

**An anecdote**: I first learned about infinite sets in high school. This was back in the day, when the internet was still a novelty to me, and I wasn't smart enough to use a pseudonym. I used to exchange puzzles with people and argue about mathematics. In one of these arguments, someone told me to read about infinite set theory. It was crazy! Infinite sets blew my mind. But by the time I got to college, they became intuitive and familiar, like a favorite old joke. Non-math people think that when math people get together, they make jokes about pi and squares. In my experience, they make jokes about infinite sets. And yet, the set of untold jokes about infinite sets remains as big as ever.

I contend that infinities are not "absurd" in the sense of "obviously false". Rather, infinities are only counter-intuitive (and only at first). Infinite set theory is well-established in mathematics. Due to some complications*, it is impossible for me to simply prove that set theory is consistent. But we think that set theory is consistent for the same reason we think arithmetic is consistent, and I don't see WLC waving his arms incredulously at arithmetic.

*See Godel's second incompleteness theorem.

This is a point that clearly needs a response, so WLC has already responded to it.

Hence, one could grant that in the conceptual realm of mathematics one can, given certain conventions and axioms, speak consistently about infinite sets of numbers, but this in no way implies that an actually infinite number of things is really possible.WLC distinguishes between "logical" possibility and "real or factual" possibility. Unfortunately, this places infinite sets in a very odd place. What kind of absurdity is this, that is too absurd for the real world, but not absurd enough for mathematics? And for what kind of "real" is it too absurd for? Lastly, if there are different degrees of absurdity, how do we know which degree it is?

There is no way to answer these questions, because "absurdity" is an intuitive idea. Either we think something sounds absurd or it doesn't. Intuition doesn't whisper in our ear, "It is absurd in reality, but not mathematics. Also, 'reality' is a category that includes past events, but not future events, and it excludes inconvenient counterexamples." Anyways, my intuition never says anything like that to me.

Put it this way. If WLC didn't know anything about what mathematicians said, he would have

*guessed*infinite sets were bad math. And then when he finds out that it's good math, what is the appropriate response? WLC's response is to say that infinite sets are still absurd, just not in mathematics. I think the proper response is to revise what we previously thought was absurd.

I must say one last thing, as a physicist. WLC thinks

*absurdity*is a good argument against a physical theory? Even when said absurdity is mathematically consistent? I question his knowledge of modern physics.

**"A few things wrong about the cosmological argument"**

1. Actual and potential infinities

2. Actual infinities in physics

3. What is real?

4. The "absurdity" of Hilbert's Hotel

5. Interlude: God is infinite

6. Forming Infinity, one by one

7. Uncertain beginnings

8. Entropy: The unsolved problem

9. Kalam as an inductive argument

10. Getting from First Cause to God

## 12 comments:

I put together an argument against Hilbert's Hotel. Let me know what your think, and how, if at all, it could be improved:

A1) Necessarily, for every x and every y, it is false that if no x is y, then some x is y.

1) If Hilbert's Hotel (HH) exists, then for any number n > 0, some rooms in HH are unoccupied if and only if the total number of occupied rooms r is (∞ - n), and no rooms in HH are unoccupied if and only if r = ∞. (Premise)

2) For any number n > 1, ∞ - n = ∞. (Premise)

3) HH exists. (Assumed Premise)

4) For any number n > 1, some rooms in HH are unoccupied if and only if the total number of occupied rooms r is (∞ - n). (1,3; MP, Simp)

5) No rooms in HH are unoccupied if and only if r = ∞. (1,3; MP, Simp)

6) Some rooms in HH are unoccupied if and only if r = ∞ - 1. (4; UI)

7) ∞ - 1 = ∞. (2; UI)

8) Some rooms in HH are unoccupied if and only if r = ∞. (6,7; ID)

9) If no rooms in HH are unoccupied, then some rooms in HH are unoccupied. (5,8; Equiv 2x, Simp 2x, HS)

10) If Hilbert's Hotel exists, then some rooms in HH are unoccupied if no rooms in HH are unoccupied. (3-9; CP)

11) It is false that some rooms in HH are unoccupied if no rooms in HH are unoccupied. (A1; UI)

::. Hilbert's Hotel doesn't exist.

Hi Godboy,

I would recommend reading about Georg Cantor and infinite set theory. Besides being relevant to your argument, it's just fun to learn about. The concept of infinity used in your argument is rather archaic, and possibly even inconsistent. It's also kind of tricky, and led to mistakes on your part.

Even if I accept your premises (which I do not), the proof does not follow. In particular, step 5 is incorrect. From premise 1 and 3, we can conclude the following:

5A) HH has unoccupied rooms iff the number of occupied rooms is (∞-n).

Or equivalently,

5B) HH does not have unoccupied rooms iff the number of occupied rooms is not (∞-n).

This looks a lot like your statement 5, except that you replaced "not (∞-n)" with "∞". According to premise 2, this is equivalent to replacing "not ∞" with "∞".

Furthermore, premise 1 appears to be incorrectly stated. HH would have unoccupied rooms if r is 0, and zero cannot be expressed in the form of (∞-n). In the case where HH has no unoccupied rooms, r is ∞, which is also equal to ∞-n.

There are also some other statements that are misstated. For example, I know what A1 is getting at, but as written it means:

[](For all classes of objects x and y, ~( (no x is y) => (some x is y) ) )

I've revised the argument. Here's the revision:

A1) Necessarily, it is false that if no x is y, then some (but not all) x is y. (Axiom)

1) If Hilbert's Hotel (HH) exists, then for any non-transfinite number n > 0, some (but not all) rooms in HH are unoccupied if and only if the total number of occupied rooms r = (aleph-0 - n), and no rooms in HH are unoccupied if and only if r = aleph-0. (Premise)

2) For any non-transfinite number n > 0, aleph-0 - n = aleph-0. (Premise)

3) HH exists. (Assumed Premise)

4) For any non-transfinite number n > 0, some (but not all) rooms in HH are unoccupied if and only if the total number of occupied rooms r = (aleph-0 - n). (1,3; MP, Simp)

5) No rooms in HH are unoccupied if and only if r = aleph-0. (1,3; MP, Simp)

6) Some (but not all) rooms in HH are unoccupied if and only if r = aleph-0 - 1. (4; UI)

7) aleph-0 - 1 = aleph-0. (2; UI)

8) Some (but not all) rooms in HH are unoccupied if and only if r = aleph-0. (6,7; ID)

9) If no rooms in HH are unoccupied, then some (but not all) rooms in HH are unoccupied. (5,8; Equiv 2x, Simp 2x, HS)

10) If Hilbert's Hotel exists, it follows that if no rooms in HH are unoccupied, then some (but not all) rooms in HH are unoccupied. (3-9; CP)

11) It is false that if no rooms in HH are unoccupied, then some (but not all) rooms in HH are unoccupied. (A1; UI, Equiv, Simp)

::. Hilbert's Hotel doesn't exist. (10,11; MT)

Hi Godboy,

I don't think your revision solved any of the problems I mentioned.

A1 is still incorrectly stated. It should say, "For all x and y, if no x is y, then it is false that some x is y." I think A1 is obvious enough that it doesn't need to be stated.

I said that your concept of infinity is archaic, but you can't fix this by just replacing every mention of ∞ and aleph-0. Aleph-0 is a substantively different concept and obeys different rules. Subtraction isn't typically defined for cardinalities.

The problem with step 5 was not fixed, it was just moved to premise 1. It basically takes the following form:

A => (B <=> C and ~B <=> C)

Where A is "HH exists", B is "HH has some unoccupied rooms" and C is "r is aleph-0". This premise does not make a whole lot of sense. I think you need to justify it.

Hi miller.

Thank you for the feedback.

My assumption is that A1 is needed in order to justify the move from (10) to the conclusion. The way you're stating it (and the way I stated it earlier), x would be both universally and existentially quantified, causing all sorts of confusion. The only way I could think to formalize your statement of A1 is as follows (let "$" be the existential quantifier):

(x)(y)($z) (~Yx => ~(z=x & Yz))

That may be true for all I know, but my revision seems to make it a lot simpler:

[] ~((x)(y)(~Yx) => ($x)(y)(Yx))

But I wish not to belabor the point. You've stated that you know what I mean and agree with what I'm saying: "No X is Y" and "Some (but not all) X is Y" are contradictory.

I'm not clear of the sense in which you feel that my idea of infinity is archaic. I understand that transfinite mathematics is lively, developing field, but my understanding is that subtraction in transfinite arithmetic is undefined only when infinity is subtracted from infinity, not when the difference is between infinity and a finite number. I've spoken to some mathematicians and they've agreed that infinity - 1 = infinity.

The first premise is "A => (B <=> C1 and ~B <=> C), where C1 = r is aleph-0 minus positive number n--not "A => (B <=> C and ~B <=> C)." Only in subsequent premises is C1 equated to C, thus bringing about the absurd consequence in (10). The justification that you are requesting is from (1) to (9), so maybe what you are really questioning is the veracity of one of the first premise's biconditionals?

I've looked "[] ~((x)(y)(~Yx) => ($x)(y)(Yx))" over and I don't think that's a proper formalization of my statement.

Tentatively, I'll bracket the issues with A1 until I can think them through further. Consider this a compliment.

Hi Godboy,

I'm not really concerned about A1, we can drop it.

I said your use of infinity was archaic, but I did not mean to say that this makes it wrong. It's just that I'm not sure what underlying assumptions are being made. This could be a problem if we were dealing with a more technical argument, but I don't think it's a problem here. We can drop this issue as well. (Oh, and I think I was mistaken about subtraction not being defined on cardinalities.)

Yes, I was questioning the veracity of premise 1. That's why I asked you to justify it. Since you advanced the argument, I think you have the burden of proof, but let me point you to where I think the problem is.

Suppose that HH exists. I can accept that if there are a finite number of unoccupied rooms, then the set of occupied rooms is equal to the entire hotel minus a finite set of rooms. So the number of occupied rooms is (∞ - n). B => C1

But does the implication go the other way? If the number of occupied rooms is (∞ - n), does that necessarily mean that there are some unoccupied rooms? The key thing here is that r = (∞ - n) does not necessarily mean that you simply fill the hotel and then remove n people. You could fill the hotel, remove n people, and then rearrange the remaining people. Or, you could simply fill the hotel, and then argue that the number of people is (∞ - n), since ∞ = (∞ - n).

Hi miller,

If we take "n" to be a placeholder for the number of unoccupied rooms, which I'll admit is not explicit in the argument and therefore renders it still incomplete, then it is inescapable that aleph-0 minus n entails there being some unoccupied rooms.

I'm working on a final draft, which will account for n qua unoccupied rooms.

"then it is inescapable that aleph-0 minus n entails there being some unoccupied rooms."

I think you need to justify that it is inescapable. You need to prove:

(There exists some positive integer n such that r = ∞ - n) => (There are some unoccupied rooms)

Or equivalently,

(All rooms are occupied) => (r does not equal ∞ - n for any positive integer n)

If all rooms are occupied, you will agree that r = ∞. You appear to be assuming that r = ∞ implies r != ∞-n for all positive n. In other words, to justify premise 1, I think you are assuming the opposite of premise 2. (Note: I think premise 2 can be easily justified with more elementary axioms.)

If you assume a statement is true in one part of a proof, and false in another part, it's not surprising that you find a contradiction.

1) Necessarily, it is false that if no x is y, then some but not all x is y.

2) If Hilbert's Hotel (HH) exists, then for any finite number u > 0 equal to the number of unoccupied rooms, some but not all of HH's rooms are unoccupied if and only if the total number of occupied rooms r = (￼aleph_0 - u).

3) If HH exists, then no rooms in HH are unoccupied if and only if r = ￼aleph_0.

4) For any finite number n > 0, ￼aleph_0- n = aleph_￼0.

5) HH exists. (AP)

6) For any finite number u > 0 equal to the number of unoccupied rooms, some but not all of HH's rooms are unoccupied if and only if the total number of occupied rooms r = (￼aleph_0 - u). (2,5; MP)

7) No rooms in HH are unoccupied if and only if r = ￼aleph_0. (3,5; MP)

8) aleph_￼0 - 1 = ￼aleph_0. (4; UI)

9) Some but not all of HH's rooms are unoccupied if and only if the total number of occupied rooms r = (aleph_￼0 - 1). (6; UI)

10) Some but not all of HH's rooms are unoccupied if and only if the total number of occupied rooms r = ￼aleph_0. (8,9; ID)

11) If no rooms in HH are unoccupied, then some but not all rooms in HH are unoccupied. (7,10; Equiv 2x, Simp 2x, HS)

12) If HH exists, then if no rooms in HH are unoccupied, then some but not all rooms in HH are unoccupied. (5-11; CP)

13) It is false that if no rooms in HH are unoccupied, then some but not all rooms in HH are unoccupied. (1; UI)

::. HH does not exist. (12,13; MT)

That's my latest, and hopefully final, revision.

Again, if "u" is specifically a placeholder for unoccupied rooms, then it necessarily follows that Hilbert's Hotel contains a number of unoccupied rooms greater than 0.

Envisage, if you could, that only rooms 4, 5 and 9 are all unoccupied; it follows, then, that HH has 3 unoccupied rooms such that r = (aleph_0 - 3). The absurdity, of course, is precisely that "No rooms are unoccupied" *does* entail that r = (aleph_0 - u), because (aleph_0 - u) = aleph_0. Hence, we obtain a contradiction.

This isn't assumed in the proof, however; it's just that given the logical trajectory that the argument takes, via premises that to my eyes are not prima facie contradictory, the contradiction is unavoidable; that's why it is a reductio ad absurdum (in the broad sense).

Hi, I don't have a good internet here, so we've reached the limit of how much I can help you with your project. My response is the same as I've been repeating. Statement 1 is still misstated, etc. Statements 2 and 5 also have syntactical errors. I note that your response to my argument that C1 does not imply B is to insist that B implies C1, so I think you missed the point there.

Anyway, good luck pinning down those syntactical errors.

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