One upside to my recent convalescence has been that I have had plenty of time for reading. Currently I'm working my way through Graham Oppy's book Philosophical Perspectives on Infinity, published by Cambridge University Press in 2006. Oppy is best known as a philosopher of religion, writing from a generally atheistic perspective. His book Arguing About Gods is really excellent, thought it definitely does not make for light reading.
As for the present book, I'm only through the first two chapters so far. I think I'm going to like the rest, though, since the preface contains remarks like this:
Part of my interest in philosophy of religion stems from the conviction that it must be possible to convince reasonable religious believers that traditional monotheistic arguments for the existence of God are worthless. Hence, not surprisingly, one of the subsidiary goals for the projected larger work is to make some contribution to the case for supposing that reasonable religious believers ought to recognise that the arguments for the existence of God provide no reason at all for reasonable nonbelievers to change their minds.
My kind of guy!
I've written about infinity a couple of times before (here and here.) But there is plenty more to say. So how about we take a quick look at one of the more intriguing thought experiments in mathematics. I refer, of course, to Hilbert's Hotel.
Let's begin with an imaginary hotel with one hundred rooms. Imagine further that all of them are occupied. If a new guest shows up in the lobby looking for a room, would we be able to accommodate him? Clearly not. At least. not without kicking out one of the guests already at the hotel.
In the context of a finite hotel the question is silly. If all the rooms are occupied and a new guest shows up, then that guest can only be accommodated at the cost of unaccommodating someone else. Seems obvious enough.
The cool part is that this is not true for an infinite hotel.
Let's imagine that we have a hotel with infinitely many rooms. The rooms are numbered sequentially 1, 2, 3, 4, ..., and for any natural number n there is a room with that number. Further assume that every one of those rooms is occupied.
If a new guest shows up wanting a room we can accommodate him by rearranging the guests who are already there. We shall simply tell the guest in room n to move to room n+1. The guest currently in room 1 will move to room 2. The guest currently in room 2 will move to room 3. The guest currently in room 3 will move to room 4.
What is the effect of this reshuffling? Well, everyone who previously had a room still has a room. No one has been thrown out of the hotel. The difference is that now there is no guest in room 1, and the new guy can be placed there.
Not a bad trick, and one that is easy to extend to any finite number of guests. If ten new guests show up, for example, then the guest in room 1 moves to room 11, the guest in room 2 moves to room 12, and more generally the guest in room n moves to room n+10. In this way everyone still has a room, but the first ten rooms have been freed up for the newbies.
We should pause for a moment to consider some practical difficulties. Obviously physics has something to say about the feasibility of such a hotel. If we're thinking of actual, physical people, in an actual, physical hotel, then we might have a problem carrying out our reshuffling in a finite amount of time. How do we transmit the message, “Move over by one room! to everyone at once? The speed of light puts a limit on how quickly a signal can be transmitted, after all.
But then, that's why we call it a thought experiment.
OK, I know what you're thinking. But what happens if an infinite number of new guests turns up, one for each natural number. Incredibly, we can still accommodate them! The trick is easily done: we simply move the guest in room n to room 2n. So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, the guest in room 3 moves to room 6 and so on. Once again, no one is thrown out of the hotel by this maneuver. Now, though, all the odd numbered rooms are now vacant, and the newbies have a place to say.
If the next night another infinite bus load shows up, we can accommodate them in exactly the same way. How about if infinitely many bus loads with infinitely many people show up all at once? No problem! We begin b vacating the odd numbered rooms as we did before. Then the first infinite bus load gets placed in the rooms whose numbers are powers of three. That is, the first new guest is placed in room 3, the second is placed in room 9, the third is placed in room 27 and so on. The people in the second bus are placed in rooms whose numbers are powers of five. In general, the guests on the n-th bus are placed in the room labeled with powers of the n odd prime. Since there are infinitely many odd primes, and since the fundamental theorem of arithmetic guarantees that no two odd prime powers can be equal to each other, we have now successfully accommodated all of the newbies.
Perhaps this seems paradoxical. We keep increasing the number of guests without making any change to the number of rooms. How can that be?
This requires some discussion, but the basic problem is in treating infinity as though it were a number. The phrase “increasing the number of guests,” which is crystal clear in the context of a finite number of guests, is not so easy to parse in the context of an infinite set.
This leads to subtle questions about how we compare the sizes of infinite sets, and to the distinction between cardinal numbers and ordinal numbers. Since this post is lone enough, we shall save that for another day. For now, let's just agree that infinity can be both weird and fascinating!
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A relevant comic, involving a dinosaur: http://www.qwantz.com/index.php?comic=1575
It doesn't discuss the solution to the problem of infinitely many infinite buses showing up, but I do like that solution to it. It's neat. And, given an infinitely large hotel, it's not really an issue that it has both infinitely many occupants and infinitely many empty rooms.
I did once contemplate how this problem could be extended to consider uncountable infinities, but the only example I came up with sounds somewhat contrived:
Suppose infinitely many people are attending the nearby math conference, and each of them has a registration number that consists of a string of infinitely many digits between 0 and 9. For each possible string there is one conference attendee, and they would all like to stay at the hotel. Can they be accomodated?
Where my breakdown with the thought experiment lies is in the assumption that an infinite hotel could be full up to begin with - it doesn't seem a coherent proposition to begin with.
But then again, doing a few undergraduate mathematics courses doesn't really equip me to be able to make that judgement. I'm still confused as to why the set of infinity containing irrational numbers is bigger than the infinity that doesn't, yet the infinity of rational numbers is no bigger than the infinity containing only integers. I suspect my mathematics professors taught me just enough to realise I know nothing about mathematics ;)
Speaking of arguments for god and infinities, are you aware of how infinities are used in the cosmological argument (by William Lane Craig in particular)? Hilbert's Hotel is frequently brought up as an example of why infinities are absurd in the real world. I always thought this was wrong-headed because if anything it's a demonstration of how our sense of what is "absurd" is incorrect.
When facing supporters of this argument, I've tried bringing up examples of infinite sets in physics, and people come up with some impressive rationalizations. One guy was insisting that the points of space aren't infinite because all points are identical.
Kel: All rational numbers can be represented by a numerator and a denominator. For argumentâs sake, the denominator can only be a positive integer, greater than zero, while the numerator can be both positive and negative.
That's three infinities to deal with: one to infinity on the bottom, and both zero to infinity and negative one to negative infinity on the top. We can use the same "free up odds" trick to fold both numerator infinities into one, and repeat it to fold the new infinity into the denominator. Easy!
Incidentally, this brings up a flaw in Jason's reasoning about the infinite busses with infinite people. We were able to easily collapse the rationals into the positive integers because there were only three infinities to deal with. Imaging counting each number as you fold it into a single infinity: positive numerator, denominator, negative numerator, denominator, positive numerator, denominator, negative numerator, and so on. Notice how it repeats? That means we'll never leave out one of the three original infinities in our sequence.
What if we had an infinite number of infinities, though? If we tried the same counting exercise as above, we will never count our way through every infinity! At best, we can only grab one value from each infinity, which exhausts our target infinity but still leaves an infinitely infinite number of values to go!
Note that *how* we interleave infinities doesn't matter. That power trick is all for naught; you cannot fit an infinite number of busses, each containing an infinite number of people, into Hilbert's Hotel.
@HJ Hornbeck: Jason's reasoning is correct. You seem to be implying that a countably infinite number of countable infinities cannot be countable, but in fact it is! However, your first counting trick with the three (or finitely many) infinities won't work. We need something more subtle; if you don't like Jason's trick (which is entirely valid) then I'll suggest another.
Label the infinite number of buses by 1, 2, 3, etc. Now label the nth occupant of bus m by the ordered pair (m,n). So the occupants of bus #4 would be (4,1), (4,2), etc. Now we must find a way to put this infinite list of infinite lists into a *single* infinite list. You said it can't be done, but here's how. First put them into a grid:
(1,1) (1,2) (1,3) (1,4) ...
(2,1) (2,2) (2,3) (2,4) ...
(3,1) (3,2) (3,3) (3,4) ...
...
You get the idea. This grid is infinite both downwards and to the right. Now start at the top left and enumerate down the *diagonals* to get the singly infinite list:
(1,1) (1,2) (2,1) (1,3) (2,2) (3,1) (1,4) (2,3) (3,2) (4,1) ...
Clearly everything in the grid appears in our list eventually since each diagonal is finite. This proves that the whole set is countable. Unload the guests from the buses in this order and they can fit into the hotel the same way a single busload can!
I've often thought that the claim (often made by some philosophers and religious apologists) that "potential" infinities do exist but "actual" ones do not, amounts mathematically to claiming that aleph-null does not exist but that C (= 2^(aleph-null)) does.
Any opinions?
To try and get the central idea across without resorting to formalism:
We say that two sets are the same size if we can match them up in a one-to-one correspondence: for each element of one there is exactly one element of the other and vice-versa.
What this means is that if there is any way of lining up some set in a counted order, like the natural numbers 1,2,3,4,... or the positive even numbers 2,4,6,8,... or the integers 0,1,-1,2,-2,3,-3,... then that set is the same size as the set of natural numbers. This size is the smallest infinite cardinal, aleph-null.
Anything that can be arranged in a discrete grid can also be arranged in a linear counted sequence: if the grid is bounded on two sides we can start at the corner and take successive diagonal lines, or if it's unbounded we can start at any point and work outward in a spiral. (It's less obvious, but still true, that we can do this for any finite number of dimensions.)
All this means that there are a lot of countable sets (all the same size): not just the naturals, integers, etc., but also the set of rationals (which we can arrange as a grid); the set of algebraic numbers, i.e. roots of polynomial equations with finite numbers of rational coefficients; the set of computable numbers, i.e. numbers which can be output by a Turing machine with a program of finite size; and so on. All of these things have the property that they, or the construct that generates them, can be ordered in a discrete counted list and therefore put in an exact one-to-one correspondence with the naturals.
The real numbers (and similar sets such as the set of all infinite sequences of integers, the set of all sets of integers, etc.) resist any arrangement into a counted sequence. For any infinite counted sequence of real numbers, we can produce a real number which is not in the list. This proves that there is no possible one-to-one correspondence between the reals and the naturals, and therefore that the set of reals is not the same size as the set of naturals. Since there are always more reals than we can accommodate in our counted sequence, we say that the set of reals is bigger than the set of naturals.
This answers Susan B's question @2; if an uncountable number of guests show up at Hilbert's Hotel, they provably cannot be accommodated (even if the hotel was initially empty): you can always prove the existence of at least one guest without a room.
And are there larger sets? Yes: one example is that the set of functions (including discontinuous functions) from reals to reals is a larger set than the set of reals itself. And so on; one can always construct larger infinite sets.
Kel,
What are the examples of actual infinities in Physics? and infinite numbers of infinitesimals don't count!
Thanks
Tom
A couple of weeks ago, the physicist Sean Carroll posted a discussion of a paper he wrote, claiming that given an infinite amount of time, the universe would be guaranteed to repeat itself. There would be another you, etc:
http://blogs.discovermagazine.com/cosmicvariance/2011/08/03/a-universe-…
I have been arguing (somewhat bumblingly) that this is not right, as an infinite path through the multidimensional state space would generally not hit every point in that space. Physical assumptions about the finiteness of the universe might ruin this reasoning, but even there, as far as I can figure, state space is countably infinite and my conclusion holds. It would be great if people who know more than I do about this would take a look at that discussion and comment there.
@miller
Craig still does use Hilbert's Hotel, but he uses another form of the problem more often. He seems to favor arguing that we couldn't traverse an actually infinite number of moments. He'll say things like, "You cannot build an infinite through successive addition," or, "You would never reach the present." There are a lot of ways you could respond, but the simplest to me seems like saying it's just a false analogy, as I argued here: http://foxholeatheism.com/fine-tuning-and-a-beginningless-past/. Wes Morriston has made similar arguments.
He also uses an argument from entropy that I actually think is a bit more compelling, but still misguided.
I suspect the architect of Hotel Hilbert, expecting this scenario, would place all the rooms on a single line - and put an infinite number of rooms _between_ each room.
Then the guests would not be put to the trouble of regular relocations (which really reduces the tips), and the only problem would be training the staff not to refer to rooms as "points", as they did back in the day when the joint was called Hotel Euclid.
darkgently: D'oh! You are indeed correct. Somehow, I never noticed it was the two-dimentional case. Guess I should brush up on my infinities...
@Andrew G.
Lane Craig identifies actual infinities with aleph-null, and potential infinities with the sideways-eight symbol. IMO, this definition is imprecise. I recently made an attempt to make it more precise. One commenter argued with the very idea of making a more precise definition, but then it's the same person who was claiming all points of space are identical.
@Tom,
I assume you meant to address me. If points in space don't "count" as actually infinite, I have no idea what rules you use to determine what counts and what doesn't. But I have more examples. In uniform cosmology, there are an actual infinite number of particles in the universe (though not all of these are visible because of the finite age of the universe and finite speed of light). Uniform cosmology could be incorrect, but I am unwilling to conclude that it is incorrect on a purely philosophical basis. Same goes for an infinite past.
"He [WLCraig] seems to favor arguing that we couldn't traverse an actually infinite number of moments." That is Zeno's Paradox. We traverse an infinite number of points every time we walk (through time or space).
Craig would say that Zeno dealt with potential infinites. I think we can learn from that, though. In Zeno's time, we had the common sense understanding to recognize that we do cross finite distances. We just didn't have the math to describe it. In our time, we have the math to deal with infinites, but we don't have the common sense experience (nor can we). I really think the "problem" is a matter of perspective. I'm not arguing that the past really was beginningless because I don't have a clue, but I do argue that it is possible.
Craig is a poopy-head. Moreover, his thinking on mathematics is not very clear. Take his paraphrased statement, "you would never reach the present." Why not, because it would take an infinite amount of time? But that was one of our starting assumptions!
Craig presents this argument as though it were reductio ad absurdum, but he never actually uncovers any contradictions, just some mental biases of his own.
This is a non-starter.For every guest N, there is a room N+1, and that room is already occupied!
Susan B: even if the [countable] Hotel was empty, you still couldn't fit uncountably many people.
The person who originated the study of infinities, Georg Cantor, was scorned not only by other mathematicians but by theologians who felt that he was encroaching on God's territory. I can't quite decipher their argument, but I guess they worried that if infinity became mathemathically discernable then God's infinite qualities would be cheapened; he would become merely infinite. Anyway, another interesting connection.
Bayesian Bouffant @ 16:
Yeah, I wondered about that myself, but I assume I must have something wrong. What's the response to this objection?
While every room is occupied by someone, they're all moving, and so getting out of each other's way. Meanwhile, there's no Room 0, so the shuffling has the effect of opening up Room 1.
I'm just beginning Oppy's Arguing about Gods, and he seems to be saying that neither atheists or theists have the arguments to convince the other to change their minds. This seems puzzling, given Oppy's own atheism, but perhaps all will be clarified. I agree that he is not easy reading.
@18: The person who originated the study of infinities, Georg Cantor, was scorned not only by other mathematicians but by theologians who felt that he was encroaching on God's territory.
IIRC, preachers were upset when Ben Franklin invented the lightning rod for similar reasons; they thought the rods prevented divine action (retribution against tall objects, I guess).
In every day and age, its the guys who yell loudest about God's omnipotence that seem to demand he needs the most help carrying out his plans.
Tom,
Why are you asking me for? I'm not a physicist. My struggle with infinity is purely conceptual.
Regards
Kel
Richard T,
the way I've heard* Oppy describe his rationale for such a statement is he didn't think that there could be an argument nonbelievers gave to believers, or vice versa, that would change their mind. That there would always be a premise that was disagreed with.
It also seems somewhat common sense, how many examples are there of people changing their mind about God from aa philosophical argument? When has the latest revision of the ontological argument made one bit of difference to the numbers of believers or nonbelievers? Julian Baggini pointed out that it's not the fine arguments that make people change their mind, and likened such arguments to cutting down a tree with a surgeon's scalpel**.
* On an episode of the podcast Conversations From The Pale Blue Dot.
** In a review of The Impossibility Of God
It's important to keep in mind that the hotel and the guests are really just literary devices in this thought experiment. If you're picturing something like the Holiday Inn only bigger then our reshuffling trick may be hard to imagine.
As CarlosT notes in comment 19, everyone is moving at once. Thus, at the same time that guest 1 is moving to room 2, the guest currently in room 2 is in the process of moving to room 3. By the time guest 1 gets to room 2 it will be vacant for him. As I alluded to in the post, you might object that it is impossible to tell infinitely many people to move all at once, owing to certain limitations derived from physics. Thus, maybe the guest in room one billion finds that room one billion one is still occupied, because that guest hasn't yet received the message that he is supposed to move over. Then we get a big pile-up and lots of angry guests. But if you're thinking like this you're being too literal.
The point of the thought experiment is to dramatize the fact that with infinite sets we can put in more stuff without changing the size of the set. Imagine that our original infinitely many guests are standing around in the lobby, each holding a slip of paper with a number on it (like the ones you get at a supermarket deli counter). Then the new guy shows up. I say, “OK everyone. Give me back your slip of paper.” They do so. I can then redistribute the slips of paper so that everyone, including the new guy, is holding one. I don't need to print up a new slip to accommodate the new guy.
For finite sets this would be ridiculous. If I have ten guests and one more shows up, then I don't have enough slips for everyone. Period, full stop.
Now, to emphasize, when I say I can redistribute the slips of paper you should not be thinking of physical slips of paper. You should be thinking instead of an abstract rule (or function) that tells me how to assign a number to any particular guest. The rule is: The new guy gets slip number 1. Every other guest is assigned a number one greater than what he started with. In this way the new guy gets a number, all my original guests still have numbers (though no one has the same number he started with) and I did not print up any new slips.
Anyway, I hope that clarifies things, but I fear I might have made things worse.
I'm with Kel on this:
"Where my breakdown with the thought experiment lies is in the assumption that an infinite hotel could be full up to begin with - it doesn't seem a coherent proposition to begin with."
Further, if you can fit more people in, it ain't full. That's just what "full" means.
In reference to Rosenhouse's statements:
"Perhaps this seems paradoxical. We keep increasing the number of guests without making any change to the number of rooms. How can that be?
This requires some discussion, but the basic problem is in treating infinity as though it were a number. The phrase âincreasing the number of guests,â which is crystal clear in the context of a finite number of guests, is not so easy to parse in the context of an infinite set."
If infinity isn't a number, there isn't a "number of guests" or a "number of rooms", we can't increase the number of either, etc. The paradox arises from treating "infinity" both as though it were a number and as though it were not.
Interestingly, "Hilbert's Hotel" becomes entirely mundane if for "infinite" we substitute "unspecified". We have an unspecified number of rooms with an unspecified number of guests, we add an unspecified number of new guests, and everyone fits. Well, OK, no problem. So, I'll tentatively conclude that "infinite" is mathese for "unspecified". :-)
aspidoscelis --
Your comment is well-taken, but I think I can clear things up. Even though infinity is not a number, it is still possible to compare the sizes of infinite sets. We can say simply that two sets have the same size if it is possible to place their elements into one-to-one correspondence with each other. That definition works just as well for infinite sets as it does for finite sets. And it is easily modified to describe what it would mean for one infinite set to have more elements than another.
Thus, I can say there are as many even numbers as there are natural numbers, even though I can't write down a familiar number that represents how many even numbers there are. I can also say that there are more real numbers than there are natural numbers. (That's requires proof, of course, but it can be done.) No matter how cleverly I try to pair up natural numbers with real numbers, I will always run out of natural numbers first.
I just noticed that how you handle the infinite number of buses, will leave a lot of empty rooms in the hotel e.g. room 15 will not be occupied. I am wondering if there is a formula that would combine bus number and seat number into a room number, so that the hotel would be full again after everyone got to their room.
(I posted a longer comment earlier that seems to have been eaten by moderation)
@27: the diagonal enumeration suggested by darkgently above can do this. The arriving guest on bus x (x in 1...) and seat y (y in 1...) can be numbered (y + (x + y - 1)(x + y - 2)/2), which numbers all arriving guests as 1,2,3,...
So just double those and subtract 1, and each arriving guest has an odd room number, while the existing guests are reassigned to the even numbered rooms.
Sorry about the moderation issues. All comments have now been liberated.
I agree.
My argument (the simple version) is as follows: if you accept that limits have some sort of real existence, then that implies that continuity also has some sort of real existence.
And once you have continuity, you have sets of cardinality 2^(aleph-null).
So it's incoherent to claim that limits exist but that aleph-null does not.
Jason @ 28:
I wish people would be more clear on the distinctions here when explaining Cantorian concepts of infinity. (Not accusing you specifically, there's just a lot of cases where jargon understood by mathematicians is casually asserted as correct in answer to non-technical questions.) Infinite sets do not have a size in the same sense as finite sets, at least as people normally understand the term. There is no number answer to the question 'how many elements are there in an infinite set'. Such a set is unbounded. Infinity is not a number, it expresses the concept of unboundedness.
Nonetheless, we can still talk about mapping from one set to another, as you say. So we can map between one infinite set and another if we have some formula/algorithm which uniquely assigns each member of one set to a member in the other. If you can do this both ways we (casually) say the infinite sets are the same size, and if you can't then one infinity is "bigger" than the other, but that is only in the one-to-one mapping sense. They are both still infinite sets, so it's a little odd to say you will ever run out of natural numbers. Rather you can show that for any explicit mapping scheme from naturals to reals, there are real numbers that don't map back.
As for what can or can't exist, I think we have to ask how we would decide if a physical Hilbert Hotel actually had an infinite number of rooms, or occupants or new guests arriving.
WEll! I'm not shopping there any more. The new guy should be at the end of the line.
No.
The line segment (0,1) contains an infinite number of points (uncountable, even) yet it is bounded and has a finite diameter (1). (This is another of those annoying errors that theistic apologists make; they assume that a past of finite duration can only contain a finite number of events, and that a past of finite duration must necessarily have a first event, and so on - none of these assumptions make any sense mathematically.)
Cardinals such as aleph-null or C are just as much "numbers" as the finite cardinals are.
@Andrew G. #32
Both steps sound vaguely incorrect to me, but it's hard to say because it is not stated precisely. I think you are already giving Lane Craig too much credit by using similar language, talking about the "real existence" of "limits" and "continuity", whatever that means. I know what a limit is, and what continuity is, but it doesn't make much sense to talk about them as objects in themselves without proper context.
I screwed up the blockquote tags above. The quote should have ended after "2^(aleph-null)."
In response to Jason Rosenhouse, #28--I share the kind of uneasiness about defining "size" in terms of correspondence that is suggested above by josh (#33).
I was also prompted me to look briefly at Cantor's "diagonal" proof that you can't map real numbers onto natural numbers. It seems very suspicious. Starting with the brief description here:
http://scidiv.bellevuecollege.edu/math/diag.html
Let's just make x the 5th element of our list of decimals. Now x5 must be unequal to itself and we've got a classic self-reference paradox. So, x can only be coherently defined if we stipulate beforehand that x is not an element in our list. Of course, if we're willing to make that stipulation, we can just stop there!
I assume this kind of objection must have been raised and answered before... but I'm no mathematician and whenever I read anything about math my mind seems to end up in some pot-hole like this.
Kell, sorry, I think I was addressing miller.
miller.
What I meant was, do we have any examples of anything that is actually infinite. Or is it always there to "make the math work". I have a similar problem to the theologians with infinity, now I am happy to accept this may just be a limitation of my perceptual framework but I haven't seen anything to convince me otherwise.
What is a point in space? Is it simply a concept to make the math work?
@Tom,
I think of a point in space as a fundamental object (and space is an infinite set of such objects, and coordinates are a mathematical construction to specify points). I dunno, it is what it is. I think all that matters is that space is a very workable idea, despite its infinite cardinality.
I think some wavefunctions (descriptions of fundamental particles) could be termed infinite. By which I mean: their non-zero probability densities extend out forever. With three caveats. One, many will have nodes of zero probability (specific points, lines, or planes on which the probability is mathematically zero). Two, probability densities will fall off to practically zero over very short distances (angstroms). Three, once they start interacting with other particles, all bets are off. A system of such particles may be finite.
It's a very amusing thought that as we scratch our heads to come up with examples of infinite objects, it could turn out that they are not just all around us, but actually more the rule than the exception. :)
@38 This is like the distinction made in programming languages such as C, between passing a structure by value or by reference. You're understanding x as a reference to a row in the table, requiring either a row number or "null". However, Cantor defined x as a value, by constructing it from its individual digits.
This hinges on a subtle but important principle of logic: If it cannot be proven that a box has been claimed to exist, then it cannot be inferred that a statement has been made about the box's contents.
In this case, the "box" is the inverse of the table's row-listing function, and its contents are the image of x under it. Because the box consists entirely of match clauses, Cantor used the trick of defining x using only not-match clauses, thereby infinitely postponing judgment on whether the box was mentioned. This invokes the principle that an infinitely recursive question is unspeakable. (Ironically, Cantor himself didn't seem to realize he was using this principle. It was stated only later by Zermelo et al.) Thus the question of whether the box can be applied to x can be discussed without objection. And the answer is, of course, found to be no.
aspidoscelis wrote: 'Further, if you can fit more people in, it ain't full. That's just what "full" means.'
I think you'll find that Jason didn't use the word "full". He used the more mathematically precise statement "every one of those rooms is occupied".
I think it helps to realise that mathematicians don't necessarily use words in the same sense as ordinary folk. They often extend or modify the ordinary sense.
I would say that, in the context of an infinite number, the expression "every one" has a meaning that's been modified. The same is true of other terms Jason uses here. They seem like familiar words, but their meaning is subtly different in the context of a discussion of infinity. That's why the conclusions seem paradoxical. They are perfectly valid when we take the terms in their extended sense, but are nonsense when we take the terms in their ordinary sense.
(The only way to fully take the terms in their ordinary sense in this discussion is to treat infinity as if it were a natural number. If we really understand that infinity is something very different, we should realise that the ordinary sense of "every one" cannot be applied to it.)
A hotel with an infinite number of rooms and a finite number of guests leads to no conundrums. Neither does a hotel with a finite number of rooms and an infinite number of guests. The complications only arise when you compare one infinity to another. And the rules for comparing infinities are not the same as for comparing numbers. This is where William Lane Craig and others go wrong, they try to fake contradictions by applying normal mathematical manipulations to infinities.
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