The absurdity of Hilbert's Hotel
Do we finally have proof that it's logically incoherent?

The absurdity of Hilbert's Hotel
jdoe25257
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Posted May 23, 2012 - 9:58 PM:
Subject: The absurdity of Hilbert's Hotel

I've been working on this argument and I want some outside opinions.

For those unfamiliar, Hilbert's Hotel is a famous thought experiment conjured by renowned mathematician David Hilbert. In it, we imagine a hotel with a countably infinite (aleph-0) number of rooms, that can accommodate up to an infinity of guests. Hence, it is said that there is a bijection between each room and each member of some countably infinite set. It seems, however, that this produces some paradoxes; for even as the occupancy reaches infinity as a limit, any new guest can be accommodated; the concierge simply needs to have everyone move one room over, and a new room will be available. Similarly, if Hilbert's Hotel is completely booked and a guest decides to check out, the occupancy remains the same at infinity.

Paradoxical though it may be, and discounting relevant nomological constraints, showing the absurdity of Hilbert's scenario is quite tricky. A healthy contingent of philosophers and mathematicians maintain that while Hilbert's imaginary hotel is counterintuitive, it is not necessarily impossible; after all, many counterintuitive things, like the behaviors of quantum particles, are clearly not impossible--why assume Hilbert's Hotel is any different? In turn, this supposedly belies the theses of philosophers, such as William Lane Craig, who maintain that concrete, actually-existing infinities are impossible.

So here, I've put together an argument from which I believe you can show that Hilbert's Hotel cannot really exist. Any feedback would be appreciated, especially on the mathematical side of things. (I'm more or less a beginner at transfinite mathematics.)


A1) Necessarily, for any x and any y, 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 number n > 0, some (but not all) 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 (but not all) 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 (but not all) rooms in HH are unoccupied if and only if r = ∞ - 1. (4; UI)

7) ∞ - 1 = ∞. (2; UI)

8) Some (but not all) rooms in HH are unoccupied if and only if r = ∞. (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, then some (but not all) rooms in HH are unoccupied if no rooms in HH are unoccupied. (3-9; CP)

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

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

Kwalish Kid
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Posted May 23, 2012 - 10:48 PM:

cgar626 wrote:
[b]1) If Hilbert's Hotel (HH) exists, then for any number n > 0, some (but not all) 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)

This just doesn't seem to make sense. There is no numeral "∞". If you want to introduce one, you'll have to give the algebraic use for this symbol.
2) For any number n > 1, ∞ - n = ∞. (Premise)

OK, so you've introduced a new set of relationships. We just have to be clear that this isn't subtraction on integer numbers.
4) For any number n > 1, some (but not all) 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)

Clearly these assumptions are inconsistent.
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Posted May 23, 2012 - 10:52 PM:

I don't think that works, since you're treating ∞ as a number (which it is not).
Tried to formalize A1 but ran into some oddities (x is defined twice as a bound variable), this may just be me though.
Will try to read in detail later on as time permits, and comment more concisely.
By the way, Craig's notion of "absurd" is different from incoherent (or logically contradictory), his notion is more vague and probably based on some kind of intuitive understanding.
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Posted May 23, 2012 - 11:05 PM:

cgar I think you're misinterpreting HIbert's Hotel. The purpose of the thought experiment is just to demonstrate that systems involving infinite numbers of things do not accord with our intuition, and it does that quite well. It's not intended to prove anything.

The fact that infinite systems are inconsistent with our intuitions should not be surprising, given that all our intuitions are formed by exposure to only finite systems.

People who try to use HH as part of a proof, such as some religious apologists, are demonstrating that they simply don't understand the thought experiment, or the concepts involved - particularly the concepts of infinity and of subtraction.

I second KK's comments by the way. Premise 1 is expressed in a way that has no meaning.

Welcome to the forums. There are lots of threads on here discussing Hilberts Hotel, and infinities more generally, if those topics interest you.

jdoe25257
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Posted May 24, 2012 - 5:44 AM:

Thanks for the feedback everyone. It seems that the general consensus is that the symbol "∞" doesn't stand for any number such that anything can be added to or subtracted from it.

So perhaps the argument could be salvaged by replacing all instances of "∞" with "0"? Aleph-0 is the number of elements in any countably infinite set, and is such that the addition of any finite set equals aleph-0. (So, for instance, if we take "2" to equal the set {ø,{ø}}, then 0 + 2 = 0.)

I probably should also stipulate that the variable n stands for any non-transfinite number > 0, since (correct me if I'm wrong) in mathematics any transfinite number--I'm supposing all of them are greater than 0--minus itself is undefined.

What do you think?



Edited by cgar626 on May 24, 2012 - 6:06 AM
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Posted May 24, 2012 - 6:36 AM:

You can't do subtraction on transfinite numbers like you can on other numbers. The rules are subtly different. As one would expect, since we can do things with transfinite sets like Hilbert Hotel. Key to this is to realize that the relationship between the total number or rooms, the number of occupied rooms, and the number of unoccupied rooms is not fixed like it is in finite sets. Really, the Hilbert Hotel example just makes use of the fact that transfinite sets have subsets of the same cardinality.
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Posted May 26, 2012 - 3:40 PM:

I've completed the final draft of my argument against the existence of Hilbert's Hotel. I've thoughtfully considered the main objections that have come up, and welcome any further comments.

First, regarding the issue of performing arithmetical operations on transfinite numbers, according to some mathematicians I've spoken to and the research I've done, it is okay to add or subtract from a transfinite; both operations are well-defined in transfinite arithmetic. What's undefined is subtracting a transfinite from a transfinite, as that could produce the following absurdity: (a) 0 + 1 = 0 + 2; (b) (0 - 0) + 1 = 2; (c) 1 = 2.

Secondly, the aleph_0 numeric, "0," indeed stands for a cardinal number, albeit a transfinite one; it's the cardinality of any countably infinite set, including the set of natural numbers, integers, odds, evens, and so on. So whatever argument one has for concluding that ∞ is not a number, this does not apply to 0.

Finally, I've redefined "n" in the original formalization to represent, specifically, the number of unoccupied rooms in Hilbert's Hotel. This means that the postulate that r = (0 - n) denotes the presence of unoccupied rooms is inescapable.

My purpose here is not in any way to vitiate transfinite arithmetic. My main thesis, rather, is that infinities applied to concrete reality lead to real contradictions, ones that wouldn't otherwise turn up in the abstract world of mathematics. Hilbert's Hotel is merely the microcosm. Here's the final draft:


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 = (0 - u).

3) If HH exists, then no rooms in HH are unoccupied if and only if r = 0.

4) For any finite number n > 0, 0- n = 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 = (0 - u). (2,5; MP)

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

8) 0 - 1 = 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 = (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 = 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)



Edited by cgar626 on May 26, 2012 - 8:35 PM
On May 26, 2012 - 6:44 PM, andrewk responded: Premise 2 is false, as Kwalish Kid has pointed out.
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Posted May 26, 2012 - 5:07 PM:

cgar626 wrote:
I've completed the final draft of my argument against the existence of Hilbert's Hotel. I've thoughtfully considered the main objections that have come up, and welcome any further comments.

First, regarding the issue of performing arithmetical operations on transfinite numbers, according to some mathematicians I've spoken to and the research I've done, it is okay to add or subtract from a transfinite; both operations are well-defined in transfinite arithmetic. What's undefined is subtracting a transfinite from a transfinite, as that could produce the following absurdity: (a) 0 + 1 = 0 + 2; (b) (0 - 0) + 1 = 2; (c) 1 = 2.

As long as you remember that you are doing transfinite arithmetic, you should be fine.
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 = (0 - u).

Well, it seems that didn't last very long. At the start of the thought experiment, there are no unoccupied rooms, but the number of occupied rooms is necessarily r = (0 - u) for any finite number u. So this premise isn't any good.
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Posted May 26, 2012 - 7:59 PM:

wolfram wrote:
"Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take one down, and pass it around, Aleph-null bottles of beer on the wall" (repeat).
grin
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Posted May 26, 2012 - 8:07 PM:

The reason why it's absurd is because you can't have a hotel with an infinite number of rooms. Everything that leads from such a ridiculous assumption is garbage.
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