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We can accomodate infinite amount of people into an Hotel with infinite rooms. Next, we can add extra infinitude of people. (See Hilbert's paradox of the Grand Hotel.) Obviously, the trick is just to postpone the solution. Instead of providing the result the method just creates an infinite (i.e. never-ending) process: you shift all people right one room, accomodate newcomer and shift the rest in the next round. I see it as infinite, never-ending process. I do not see what you get in result. I do not see infinity-hotel accomodating 2 infinities of individuals. We say: ok, we will never terminate. That is why we can make accomodate 2n in the Hotel of size n. Good cheat.

Cantor says that we can enumerate all sets of integers but this list of sets will be incomplete because we can always scan it and build another set, Δ, which is different from all the sets in the list. Despite the process does not terminate either, the math community yet consider the enumeration of Δ impossible.

I see that it would be an absurd if you could accomplish building the Cantor antidiagonal set. But, accomodating 2n people into n rooms is identically absurdish. What makes the Hilbert conclusion less absurdish? Why cannot we be happy with existence of antidiagonal, which contradicts to the common sense? The process of diagonalization is never ending, as is the acommodation of guests in the Grand Hotel anyway.

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    specifically, I'm having some trouble seeing how this is a genuine question. It seems more like a platform rant or something like that. – virmaior May 11 '14 at 14:51
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    There is a nice paper by Wilfrid Hodges on the surprising fact that time and again people come up with failing refutations of Cantor's diagonal argument: W. Hodges: 'An editor recalls some hopeless papers'. Bulletin of Symbolic Logic 4 (1):1-16 (1998). – sequitur May 11 '14 at 14:55
  • @Val: I edited your question to improve it. If you feel I misinterpreted your statements, please feel free to change my edits. – DBK May 11 '14 at 15:46
  • It's not a never-ending process. You tell the person currently in room n to go to room 2n, and then the new infinity of guests can stay in the odd-numbered rooms. – sjmc May 11 '14 at 15:47
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    @sequitur: We should consider ourselves lucky that the number of people publishing failing refutations of the diagonal argument is apparently only countable infinite. – gnasher729 Apr 3 '16 at 16:03
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1) I see it as infinite, never-ending process.

YES, It is : every infinite process is never-ending.

2) I do not see what you get in result. I do not see infinity-hotel accomodating 2 infinities of individuals.

With an infinite Hotel it happens ... Two "infinite" set (if we assume that here infinite stay for countable) "added together" gave us a new "infinite" set which is still countable, i.e.it includes the initial sets but has "the same number" of elements (in a precise mathematicals ense of "having the same number of elements").

3) we can make accomodate 2n in the Hotel of size n.

NO; in order to accomodate 2n people we need the first 2n rooms... of course. But the Hotel has an infinite number of rooms; thus, for every finite n, we have always rooms enough for n people, and also for 2n of them, and also for 3n of them ...

3

The way to accommodate infinitely many people at once is not to accommodate them one at a time. Rather, you ask everyone to look at their room number, and then move to the room with twice that number. Now you've got infinitely many rooms free (namely all odd-numbered), where you can put those infinitely many extra guests.

Note that the moving is not a sequential process, the people can change their rooms all at the same time. Of course, to make that arrangement possible, the rooms need to be arranged that the time needed to change from room n to room 2n is bounded. Or maybe, you just have to move with a speed proportional to n; people who live in infinite hotels are able to move arbitrarily fast. ;-)

Now, if two infinite groups arrive at the same time, you just ask the first arriving group to take every second of the odd-numbered rooms, and thus there are still infinitely many rooms free for the second arriving group.

2

Would Cantor's results be more palatable if they were stated thusly:

IF you accept the axioms of Zermelo-Fraenkel (ZF) set theory and the rules of basic logic;

THEN Cantor's results are true.

Nobody is saying these things are true "in real life" or in the physical universe. The Hilton corporation doesn't own a hotel that has infinitely many rooms. Rather, these results follow from logic, once you assume ZF. Hilbert's hotel is just a story, a fable whose purpose is to illustrate the ideas of bijection and cardinality. There really isn't any such hotel in the world.

One is free to disbelieve the Axiom of Infinity, which gives us infinite sets. In fact it's perfectly sensible to consider ZF-, which is ZF with the Axiom of Infinity negated. This is a consistent system in which the collection of natural numbers is not a set; nor is there a set of real numbers.

Would this be more satisfying to skeptics? Then they could aim their objections at the true culprit: The Axiom of Infinity, which says that there is an infinite set. This is manifestly false in the everyday world. So the skeptics could simply accept Cantor's results as a fallacy originating from the acceptance of infinite sets. And we could answer them plainly: Fine, no problem. You're a finitist. Nothing wrong with that, it's a logically sound position. We have no fundamental disagreement. I accept a particular axiom, you reject it. There's no right or wrong here.

Would this satisfy the OP?

Some of the information contained in this post requires additional references. Please edit to add citations to reliable sources that support the assertions made here. Unsourced material may be disputed or deleted.

  • It seems that you say that there is no answer. I do not understand your attitude, however. Why do you think that I am against Canot's method but not against Hilber's one? Are you a bigot? – Val May 11 '14 at 19:22
  • @Val and user4894. Try to be nice. Both of you. – user3164 May 11 '14 at 19:24
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    @Watson, Can you explain why I should "try to be nice?" Can you explain where I have not been nice? I posted a very sensible response to this frequently asked question. – user4894 May 11 '14 at 20:53
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    @Val, What did I say to offend you? You expressed the idea that these infinitary results are "absurd, your word. I replied that these results are neither absurd nor surd, or whatever the opposite of absurd is. Rather, they simply follow logically from some axioms. Pick different axioms, get different theorems. Why do people think I said something objectionable? I'm really baffled here. – user4894 May 11 '14 at 21:01
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A simpler example of the phenomenon you are maybe trying to grasp is that of a recursive definition: for example, the Fibonacci numbers are a sequence of natural numbers defined by the three equations

  • F0 = 0
  • F1 = 1
  • Fn+2 = Fn + Fn+1

If it suits you, you can think of this as defining a 'never-ending process' for continually generating successive values of F. Just be sure not to confuse yourself — you must still also grasp that the entire sequence has been completely defined simply by writing down these three lines.

The procedure you describe in the opening post is similar. Ultimately, you are inductively defining two functions f and g on the natural numbers:

  • f(n) tells you the room where the person who started in room n ultimately moves to
  • g(n) tells you where newcomer number n will ultimately reside

The important point of the "never ending process" you describe is that each individual person eventually settles down to a specific room after a finite number of steps and is never disturbed again; this ultimate location is the value of the corresponding function.

Your post is a little vague — the procedure you have in mind may, in fact, not have this property, although treating that is beyond the scope of this post. I will, however, give a precise description that does have this property:

On round n, the people in rooms 2n and up all shift over one room, and newcomer n takes room 2n.

(note that I start my indexing at zero) You can formulate and solve the recursion to determine

f(n) = 2n+1 and g(n) = 2n

In other words, the end result is just to interleave the current residents with the newcomers.

Since the Hilbert Grand Hotel already has a room for each natural number, we see that no new rooms were built, each original resident occupies a room and each newcomer also occupies a room. (and no room contains more than one person)

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You are mistaking popularisations of mathematics for mathematics. The mathematical formulations of the results in question talk about functions, not processes. The functions are “just there” in the same way as the set of natural numbers.

Since your question arises from this misconception, there is really not much else to say about this.

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Hilbert's conclusion contradicts Cantor's. Absurd is only that neither Hilbert nor the actual set theorists have understood that fact.

Cantor's diagonal procedure is based on the claim that the infinite sequence N can be completed, such that the set of real numbers in his "list" is well-defined and no further number can be added. Of course that is nonsense; it is impossible to complete an infinite sequence. Therefore Hilbert and everybody who refuses the existence of a finished infinite sequence or list or hotel is right. And Cantor's argument breaks down.

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    Cantor's argument is a reductio ad absurdum, or proof by contradiction. First we suppose for the sake of argument that the real numbers can be enumerated (arranged in a countable sequence). Then we show that this leads to a contradiction: there's at least one real number that can't have any place in this sequence. The conclusion is that we reject the supposition. So Cantor doesn't assume that the real numbers can be enumerated. The conclusion of the proof is that they can't be enumerated. – Dan Hicks Jul 16 '17 at 10:57
  • Cantor assumes that all counting numbers can be used to construct a list. That is nonsense. For all counting numbers we can prove that they are not all counting numbers. Sounds odd? But is fact! Every counting number that can be listed belongs to a finite initial segment and is followed by infinitely many counting numbers. – Heinrich Aug 25 '17 at 20:12
  • When mathematicians speak of (infinite) lists: they usually mean a function whose domain is the natural numbers (or some equivalent notion). A list containing all "counting numbers" would mean a surjective function from the natural numbers to the counting numbers. An example of such a thing would be the identity function (assuming you use "counting number" to mean "natural number"). – Hurkyl Dec 13 '17 at 10:38
  • @Hurkyl: When mathematicians speak of all real numbers they (should) know that there are not all real numbers. Same holds for the natural numbers as far as they can be applied in mathematics. It is impossible to define a natural number that has Kolmogorov-complexity of 10^80 or more. The identity function maps the typical natural number n on itself. The belief in "all natural numbers" as the domain of this mapping is extremely naive. – Wilhelm Dec 13 '17 at 14:06

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