The "parable" of Hilbert's Hotel (NOT the paradox)
This story presents an original argument against [math]0.{\overline{9}}=1[/math]. Please let me know if there's a flaw in this story.
The Story:
The Potentially Infinite Hotel is continuously growing but always finite. Since it is growing, new rooms are always being added so there is always room for more guests. When a new guest comes, the hotel manager always shifts each guest up one room, freeing up the first room for the new guest.
One peculiarity about this hotel manager is that he's a numbers enthusiast. So when he wants people to move rooms, all he does is announce a number on the PA system and everyone is expected to know exactly what to do.
Here's how his number system works:
He keeps track of the occupancy state of the entire hotel with a single number: the occupancy number. Each digit describes the occupancy state of a different room whereby a 0-digit indicates that the room is vacant, while a 9-digit indicates occupied.
This system works perfectly for him for every N.
But one day his dream comes true and he gets hired at Hilbert's (actually infinite) Hotel and since his numbers system worked perfectly at the Potentially Infinite Hotel, he decides to use it at the fully occupied Actually infinite hotel.
A new guest enters and as usual the manager wants everyone to shift up one room. Because there are infinite rooms (i.e. N = infinity), he jots [math]0.8{\overline{9}}1[/math] on a piece of paper but before he gets a chance to announce that number the new guest points out the following:
And so the hotel manager announces 0.9 on the PA system. As bolded above, the guest from room 1 vacates his room and frustratingly stands out in the hallway as he watches the new guest happily take his room. Unlike the paradox, nothing magical has happened in this story. After all, you can't get something from nothing.
The hotel manager quits his job and returns back to the Potentially Infinite Hotel. At least his number system works there.
What if we do the same? What if [math]0.{\overline{9}}[/math] is not a number with actually infinite digits but instead shorthand for an endless operation: 0.9 + 0.09 + 0.009 + ...?
The Story:
The Potentially Infinite Hotel is continuously growing but always finite. Since it is growing, new rooms are always being added so there is always room for more guests. When a new guest comes, the hotel manager always shifts each guest up one room, freeing up the first room for the new guest.
One peculiarity about this hotel manager is that he's a numbers enthusiast. So when he wants people to move rooms, all he does is announce a number on the PA system and everyone is expected to know exactly what to do.
Here's how his number system works:
He keeps track of the occupancy state of the entire hotel with a single number: the occupancy number. Each digit describes the occupancy state of a different room whereby a 0-digit indicates that the room is vacant, while a 9-digit indicates occupied.
- If he wants to vacate room 1, he needs to subtract 0.9 from the occupancy number so he announces 0.9 on the PA system.
- If he wants to vacate room 2, he needs to subtract 0.09 from the occupancy number so he announces 0.09.
- If only room 1 is occupied and he wants to shift that guest up one room, he needs to subtract 0.9 (to vacate room 1) and add 0.09 (to occupy room 2). In other words he needs to subtract 0.81 so he announces 0.81.
- If there are only guests in rooms 1 and 2 and he wants to shift them up one room, he announces 0.891.
- In general, if there are N guests in room 1 to N and he wants to shift them up one room, he announces 0.8 followed by (N-1) 9's followed by a 1. For example, if N=5, he announces 0.899991.
This system works perfectly for him for every N.
But one day his dream comes true and he gets hired at Hilbert's (actually infinite) Hotel and since his numbers system worked perfectly at the Potentially Infinite Hotel, he decides to use it at the fully occupied Actually infinite hotel.
A new guest enters and as usual the manager wants everyone to shift up one room. Because there are infinite rooms (i.e. N = infinity), he jots [math]0.8{\overline{9}}1[/math] on a piece of paper but before he gets a chance to announce that number the new guest points out the following:
- [math]0.0{\overline{0}}1[/math] is vanishingly small, so small that you cannot name a number between [math]0.0{\overline{0}}1[/math] and [math]0[/math] so the two are equal. [math]0.0{\overline{0}}1 = 0[/math] and therefore [math]0.8{\overline{9}}1 = 0.8{\overline{9}} [/math]
- Furthermore, the difference between [math]0.8{\overline{9}} [/math] and [math]0.9[/math] is vanishingly small, so small that you cannot name a number between them so they too are equal. [math]0.8{\overline{9}} = 0.9 [/math]
- Therefore instead of saying [math]0.8{\overline{9}}1[/math] he should simply say [math]0.9[/math].
And so the hotel manager announces 0.9 on the PA system. As bolded above, the guest from room 1 vacates his room and frustratingly stands out in the hallway as he watches the new guest happily take his room. Unlike the paradox, nothing magical has happened in this story. After all, you can't get something from nothing.
The hotel manager quits his job and returns back to the Potentially Infinite Hotel. At least his number system works there.
What if we do the same? What if [math]0.{\overline{9}}[/math] is not a number with actually infinite digits but instead shorthand for an endless operation: 0.9 + 0.09 + 0.009 + ...?
Comments (104)
I'll use '*' instead of the overbar.
0.89*1
is not defined.
There is no real number that has an infinite decimal expansion but with a final entry.
Your imaginistic scenario, not even itself approaching a mathematical argument, not even of alternative mathematics, is done. Argument by undefined symbolism is a non-starter.
You are typical of cranks who argue with undefined terminology and symbolisms. Using terminology and symbolisms in merely impressionistic ways.
/
Quoting keystone
On your own finitistic terms, at any point, the sequence is finite. 'continuously growing' is never witnessed. Only finitely many individual finite sequences.
0.938K
330.five
"integer division by Olaf Gerdmuller"
@god must be atheist
While I think my notation of putting a digit after the repeating term is an interesting way of potentially representing an infinitesimal, I understand why you might not like it. However, your criticism of it doesn't hurt the overall argument. I've rewritten the story below, but in binary where I don't have to resort to putting anything after the repeating term.
-------------------------------------
Here's how his number system works:
He keeps track of the occupancy state of the entire hotel with a single binary number: the occupancy number. Each digit describes the occupancy state of a different room whereby a 0-digit indicates that the room is vacant, while a 1-digit indicates occupied.
If he wants to vacate room 1, he needs to subtract 0.1 from the occupancy number so he announces 0.1 on the PA system.
If he wants to vacate room 2, he needs to subtract 0.01 from the occupancy number so he announces 0.01.
If only room 1 is occupied and he wants to shift that guest up one room, he needs to subtract 0.1 (to vacate room 1) and add 0.01 (to occupy room 2). In other words he needs to subtract 0.01 so he announces 0.01.
If there are only guests in rooms 1 and 2 and he wants to shift them up one room, he announces 0.011.
In general, if there are N guests in room 1 to N and he wants to shift them up one room, he announces 0.0 followed by N 1's. For example, if N=5, he announces 0.011111.
This system works perfectly for him for every N.
But one day his dream comes true and he gets hired at Hilbert's (actually infinite) Hotel and since his numbers system worked perfectly at the Potentially Infinite Hotel, he decides to use it at the fully occupied Actually infinite hotel.
A new guest enters and as usual the manager wants everyone to shift up one room. Because there are infinite rooms (i.e. N = infinity), he jots 0.01 (where underline is used here to represent repeating) on a piece of paper but before he gets a chance to announce that number the new guest points out the following:
The difference between 0.01 and 0.1 is vanishingly small, so small that you cannot name a number between them so they are equal. 0.01 = 0.1. Therefore instead of saying 0.01 he should simply say 0.1.
And so the hotel manager announces 0.1 on the PA system. As bolded above, the guest from room 1 vacates his room and frustratingly stands out in the hallway as he watches the new guest happily take his room. Unlike the paradox, nothing magical has happened in this story. After all, you can't get something from nothing.
The hotel manager quits his job and returns back to the Potentially Infinite Hotel. At least his number system works there.
Stick with decimal for my response. If the occupancy number is 0.9909 and he wants to shift the guests in room 1 and 2 up, he needs to perform an operation on the occupancy number to make it 0.0999. The difference between these two numbers is 0.891 so that's the number he would announce. Does this make sense to you?
Hopefully the binary version of my story can put your concerns about undefined terminology and symbolism aside.
Have you understood my story and disagree with it or are you criticizing it without understanding it?
Are you saying that nobody is in a position to say that a process goes without end because they cannot witness the end?
I noted the first place in your original post that you spouted nonsense. Now you've revised. If I'm in the mood, I'll give you a second chance.
Quoting keystone
That is more nonsense.
Just dont let guys like Tones put you off as you can see hes clearly the moody type but generous depending on the lunar cycle.
As for the problem posed by Hilbert himself I think its worth noting that his playfulness with the concepts of infinity or infinates is not something new but he has clearly presented mathematicians with a carefully constructed brick wall.
You're trying to use "announcing numbers" to stand for two different things : emptying a room into the hallway and shifting occupants to successive rooms. It can't be both.
Under your scheme, announcing 0.9 creates the same problem for a finite hotel as an infinite hotel : any occupant of room 1 is now standing in the hall !
Proof that 0.891 = 0.9 : announcing either 0.891 or 0.9 leaves the infinite hotel in an identical state, namely 0.09
There's only one.
Then I shall have to clone you an infinite amount of times to satisfy hilberts problem.
Please sign below
Wow, you're way off base. See my reply to keystone, above.
Again,
0.9 - 0.89 = 0.09 (You don't need 1 on the end)
and
0.9 - 0.9 = 0.09
So, 0.89 = 0.9
I am confused. At what point did I offer you any math for such a refutal ?
Genius.
Or to remove the decimal 9-9 = 0.9
You have truly distorted and confounded my conception of Math.
Next on the curriculum
1 + 1 = -5 or 3 or anything apart from 2
What "tweaking" of keystone's incorrect approach "solves" the problem?
Also, claiming Hilbert constructed a wall for mathematics is a strange take.
Essentially the use of non-linear equations could be useful.
A strange take indeed but thats what math is strange!
I'm using an underscore as the symbol to represent an infinitely repeated digit, as keystone has. Try reading it again.
Example, please.
Sure, the simple application of a non-linear equation solves the countable infinity problem that was proved by Cantor and consequently the first argument of Hilbert in his presentation of the above paradox.
So where's the wall?
A redundancy in terms by applying Occams Razor
And I still don't see how you get from keystone's linear approach (numbers are only added or subtracted) to non-linear systems. Seems like a fair amount of "tweaking".
You think the phrase "infinity of infinities" is suspect? How so? Use math please.
Infinity + Infinity + infinity ? = ? Infinity
What specifically do you refer to ?
Let me help : There are an infinite number of rational numbers between 0 and 1. Also between 1 and 2. And so on. An infinity of infinities. And yet Cantor proved that the cardinality of the set of rationals is the same as the cardinality of the integers - they can be placed in a 1-to-1 relationship.
No problem there.
In ordinary set theoretic context, there is no object called 'infinity' that is an operand in an addition operation.
Quoting Deus
Is that supposed to be the leminscate?
Then this is also not new but the old re-formulation of Xenos Paradox
Thus we have found problems in such a framework.
(Using the apostrophe for '...')
for any x an y,
x'y
doesn't stand for an real number expansion.
Okay, I get it. You're trolling.
As with other debates we've had on TPF, we find philosophy majors trying to bring the wishy-washy terminology of philosophy into a math discussion. Infinity is not understood as a math concept, but as a poorly defined common-language concept. :roll:
I am not. Set theory is self-limiting in the way the theory is formulated.
Define 'self-limiting theory'.
Yeah, I couldn't figure out how to indicate repeated digits. The underscore also seems to work if we agree on usage.
Which one? Be specific.
(Hmm, coming up with your own definition of infinity ... is this MU ?)
Dichotomy Paradox
'flexible enough', 'deal with operations', 'render useless'.
Not even philosophical, let alone mathematical.
Deus is a crank troller.
Happy for any label at this point. However as a logician yourself give yourself the imaginative prowes to scrutinise the above statement and where proof is warranted provide it.
I could simply call you an unimaginative logical troll by the same token.
But I wont.
Oh, I see. So you don't believe that an infinite set can be dense. Got it.
We speak different languages.
(And I gotta hit the hay. TIDF, have fun.)
Also tones your custom way of insulting is to belittle the other guy by calling them either troll or crank but here youve basketed the two together.
The point is not to insult, but rather to flag the situation. Usually cranks are not trolls, since they are sincere, though horribly self-misguided. But I take you as a troll since you don't even offer arguments but just simple flippant nonsense.
I understand you better. This then appears to be crank to crank communication at its finest.
https://www.memedroid.com/memes/detail/3284903/Clown-to-clown-communication
This is unbecoming of a Logician Tones. First you called me a crank troll but in the above statement Im only referred to as troll.
Which is it ?
A teetotaler can't be a drunktroller.
Quoting Deus
I'm not.
Both.
False premise.
Premise presumes to know about the subjects drinking habits.
Who is the crank now ?
I didn't know whether you're talking about yourself or about me. (And I couldn't resist the wordplay.) In any case, if your point is that you're better at least for not being drunk, then congrats.
Fair nuff.
Quoting TonesInDeepFreeze
I agree that as a number it's nonsense...but I believe so are infinitesimals. This is the whole essence of my argument after all. I'm not going to defend infinitesimals nor am I interested in debating the validity of non-standard analysis. But I can certainly write a program to output digits corresponding to 0.891. It's just that that program can never be executed to completion so it would never reach a moment where it would output a 1 digit.
Im talking about the premise:
A teatotaller <> drunktroller
Your mistake in coming up with the above argument was to presume the subject was drunk from the predicate.
Infinitesimals are rigorously handled in non-standard analysis. It's not a question of validity.
I don't know what '<>' is meant to symbolize. If it is for some form of equivalence, it's the opposite of what I said. No teetotaler is drunk.
<> means does not equal. A revaluation of facts and reality is required on your part
Please back this up before I declare it a nonsensical flippant assertion.
You think it is incorrect to say that no teetotaler is drunk?
Refer to the subject of non-standard analysis in mathematical logic (starting with Abraham Robinson), or, with a different method, internal set theory. You may consult many a book or article. Probably many articles on online.
Thanks for your kind words. Tones has been generous with his time to me in the past so I can't complain, but his tone does sometimes hurt a little. (woe is me :P)
I didn't assume anyone is drunk.
Consider the statement
All bachelors are unmarried men.
That is watertight and contains truth as its self referential.
Do you claim that teatotallers are not drunktrollers ?
If so I am ready to destroy the above argument.
The word is 'teetotaler'. No teetotaler is drunk, therefore no teetotaler is both drunk and a troller.
Then there will exemptions to the first statement.
There could exist such a place where a teetotaller abstains from drinking on all occasions apart from when he unknowingly drinks alcohol because his wife cant be trusted.
This then proves that a teetotaller is drunk and it follows that also a troller.
Your logic has been found wanting.
Whatever you have in mind, it's not a program. If P is a program to print the entries in a denumerable sequence, then for each entry, there is step at which that entry is printed.
As in other threads, you're using technical sounding verbiage without regard for making sense with it.
Hes referring to the halting problem in relation to turings complete machine
Depends on the exact interpretation of a given definition. If taken literally in the sense of 'practices complete abstinence' then drinking alcohol even inadvertently makes one no longer a teetotaler at that moment. But I grant that ordinarily, probably most people wouldn't regard that as failure to maintain being a teetotaler; and I overlooked that possible situation. So, I'll give you not a full point for that one, but at least most of a point.
A pint did you say ? I will happily accept a Guiness
I'm referring to a computer program. For example,
N=1
print(0.8)
while N+1: # see Comment 1 below
{ print(9)
N+=1
}
print(1)
# Comment 1: This is assuming that once you reach the largest possible number on the computer it returns 0 (effectively going full circle), which breaks the loop and prints a 1.
On an infinite computer it will never print a 1, but I can still write the program.
No he's not. The halting problem is not that there are programs that don't halt. But rather that there is no program to decide whether any given program and input will halt.
You don't know what you're talking about. You're just throwing out red herrings. A form of trolling.
It was an inference Tones, albeit an implied one at that. I saw where keystone was getting at.
If I am throwing red herrings its because i have surplus fish and you have earned yourself a fish. Its got omega 3 and good for you.
And most certainly would make you less cranky and grumpy.
You can write whatever you like, but my point, as seen in context, is that it's not a program to print all the entries in the sequence.
Consider the ascending sequence of members of w+1 (omega plus one). That is a denumerable sequence with a last entry. But there is no program to write all the entries in that ascending order. On the other hand, it's trivial to have a program write:
1, 1/2, 1/4 ...
while 0 is not an output.
That's just a starker example of what you're doing. Yes, it's a program, and it outputs every successive halving. But 1 is not an output of the program.
In the infinite hotel 0.89 = 0.9. Therefore shifting everyone up one room is equivalent to vacating room 1. And since vacating room 1 doesn't create more empty rooms, one should be suspicious about what is achieved in shifting everyone up one room.
In the finite hotel every number has a unique instruction. 0.9 means only one thing: vacate room 1. Infinite decimals are not required for the finite hotel.
Take the instance a program that aims to output an infinate number such as Pi it will run and not halt.
However with well defined input paramters the problem of it is easily solved.
I was just reading this thread, but it seems you have solved your own conundrum. In the infinite hotel the two are equivalent, as you yourself point out. So 0.9 recurring is equal to 1.
And in the finite hotel they are not equivalent, as you point out. So 0.9... with 9 repeated a finite number of times is not equal to 1.
I understand your argument, but I think you didn't read my code (especially the commented line) so you don't realize that it can be run on any finite computer and when doing so it will always eventually output a 1. The larger the computer, the later it will output a 1 but that 1 is unavoidable. Of course, if the code were run on an infinite computer the program would never halt and therefore the 1 never comes.
Then it will miss outputting one of the 9s.
You can't have cake and eat it too.
If it runs only finitely many steps but outputs the 1, then it skips an infinite number of the 9s.
If it runs without end, then it outputs each of the 9s, but never outputs the 1.
Right, and my conclusion is that the number system which was developed using finite intuitions breaks down when extended to infinity (the infinite hotel). And I want to suggest that this may be what's happening with math. The number system using numbers which was developed using finite intuitions (rational numbers) breaks down when extended to model the continuum (with real numbers).
You don't like that mathematics for the sciences doesn't comport with your understanding of impossible fictional realms. Yeah, that's a real dagger in the heart of the mathematics for the sciences.
Think of the counter within my program like an odometer that starts at 1 and has 5 digits. Eventually you will reach the 99,999 after which it increments up to 00000. At that point the program prints a 1.
The program executed on a finite computer will print a 1 and it will print it too early. I agree with you on this. But run it on a bigger computer and it will do a better job, printing the 1 after more 9's. Take it further and run it on increasingly bigger computers and get increasingly better approximations. But can you go the limit and run it to completion on an infinite computer? No. Nor do I believe that an infinite computer exists. But as someone who is fond of limits, I would have expected you to appreciate that the program has value even though we cannot literally go the limit.
I'm working in Hilbert's fictional realm. Are you saying that it's impossible? Also you're not in a position to critique my understanding of that realm since you haven't even tried to understand my opening argument.
Mathematics for the sciences? Try floating-point numbers. We don't use real numbers.
Printing it early is no trick.
Quoting keystone
".89[...]" is notation for a limit. And that limit is .9. And ".9[...]' is also notation for a limit. .9[...] = 1.
Set theory is abstract. It doesn't have hotels. To be more exact, I should say that from an imaginary analogy to set theory, you impose an incoherent interpretation. It's incoherent because you start out by describing a program to output values (presumably in a certain order) but it's not a program.
I exhausted loads of my time and patience with Thomson's lamp with you. You're making a variation of the same mistake here.
Yes, from the axioms of set theory, we derive the theorems of calculus.
Ridiculous.
Calculus was developed well before set theory came into the scene. Also in the field of mathematics its nothing more than a minor development/distraction
Quoting keystone
And that's exactly what the hotel manager concluded in my story. Your criticism of my story was of an inconsequential intermediate step. And even now, you focusing on the program is secondary. You want to argue without listening to my original argument.
The program isn't even a part of the story and the repeating term followed by a 1 was just an inconsequential intermediate step that was just an artifact of the decimal system. It's like you're criticizing a written argument by saying 'this i has not been dotted and this t has not been crossed so this paragraph makes absolutely no sense'.
If you see no value in thought experiments then you are without a very important tool.
Quoting TonesInDeepFreeze
It's you who's repeating the past by not listening to my initial argument. Instead you're nitpicking.
But that is not what is happening here. Using finite intuition would not lead to thinking 0.9... = 1. So the maths that demonstrates 0.9... = 1 is not using finite intuition.
However it is you who is trying to analyze it using finite intuition, which is the source of confusion I think.
Besides, 0.9... is a rational number, so I don't understand your last sentence in this instance.
Let me explain in detail why your scheme is flawed. I'll only use a minimum amount of math, promise.
The hotel manager uses the technique of "announcing numbers" for two entirely different reasons. Sometimes, announcing a number removes an occupant from a room and does not specify where they go. Other times, announcing a number shifts occupants to successive rooms. But, imporantly, these algorithms are not the same.
But it's worse than that. Each of these algorithms only works for a tiny set of numbers. Consider a hotel with occupants in just the first two rooms (0.99). Only two rational numbers can be announced which shift occupants to new rooms : 0.891 and 0.081. And only three rational numbers remove occupants without shifting : 0.9, 0.09, and 0.99. But the set of rational numbers is infinite, so this cannot be the operation of subtraction defined for the rationals. For example, announcing 0.783 is meaningless, but subtracting 0.783 is perfectly valid.
You have mistaken the manager's actions for subtraction because your technique of announcing numbers just happens to give the same results as subtraction in these 5 cases. And you still had to give different processes for 0.891 and 0.081 on the one hand and 0.9, 0.09, and 0.99 on the other to force the outcomes to match subtraction.
In order for your technique to correspond with subtraction, you would need to describe a single algorithm that could handle all rational inputs. And then show a contradiction.
Crucially, announcing 0.89[...] and announcing 0.9 in the infinite hotel are NOT identical. One shifts occupants, the other removes the occupant in room 1 without specifying their destination. Interestingly, both leave the hotel in the same state : 0.09[...]
Remember, a broken watch tells the correct time twice a day.
It's about taking finite intuitions to the limit. In the hotel story, he takes his system which works with finite rooms and applies it to infinite rooms. In math, we take our intuitions developed from numbers with finite digits and apply it to numbers with infinite digits.
Maybe this will help communicate my view: The fact that in decimal notation 1/3 has infinite digits is a red herring. 1/3 on the Stern-Brocot tree is represented as LL because we go Left-Left from 1/1. Every positive rational number can be represented on this tree using finite characters/digits. 1 is [] since it's the starting point. If we look at the numbers of LR, LRR, LRRR, LRRR, LRRRR, etc. we see that we're listing numbers that approach 1 and if we keep going we have LR. This corresponds to 0.9. IF we were to go the the limit and hit the bottom of this tree then LR=[] (i.e. 0.9=1), but there is no bottom of the tree, so they are not equal. LR is not a number, it describes an endless journey/process down the tree. Similarly, 0.9 is not a number, it represents an endless operation [0.9+0.09+0.009+...].
The reason why you think 0.9 is rational is because you believe it equals 1, which is indeed a rational number.
Firstly, thank you. It is clear that you understand how the hotel manager's number system works and I really appreciate that!
You're right that there are a lot of announcing numbers that don't have meaning for him (e.g. 0.783). I started with decimal notation because of convenience/familiarity but have faced much criticism for the artifacts it introduced such as the idea of 0.001 being a number. Soon after my initial post I reframed it in binary to address these issues. In binary, every number between 0 and 1 has meaning in the hotel manager's number system. And just as we can show that there's a bijection between (0,1) and (-inf, inf), we can develop a system whereby every real number has meaning in the hotel manager's number system. Roughly speaking, here are the steps:
1) You provide any real number
2) I convert it to binary
3) Using the bijection, I find the correspond number within the range (0,1)
4) That number has a meaning in the hotel manager's system
You are unnecessarily confusing yourself. 0.9... IS a rational number. It is not that I think it is, rather it is.
Any number that infinitely repeats a finite sequence after the decimal point is a rational number. 0.9... repeats the finite sequence "9" infinitely, so is a rational number.
Irrational numbers, like Pi or the roots, don't have finite sequences repeating infinitely.
Switching to binary was an inspired choice. Now in the infinite hotel, every value between 0 and 1 has meaning when announced by the manager (though that is not true for any of the finite hotels).
But now consider the state of the infinite hotel after either 0.1 or 0.01[...] are announced. If the infinite hotel starts out with every room filled, in both cases, room 1 will be empty, and all the other rooms (all [math]\infty[/math] of them) will be occupied. That is to say, announcing 0.1 or announcing 0.01[...] have the same effect on the hotel. They are equivalent as far as the hotel is concerned. (I.e., 0.1 = 0.01[...] in your prposed scheme)
In both cases, the occupant of room 1 must leave room 1. In the case of 0.01[...], they then move into room 2 (of course bumping everybody up), and in the case of 0.1, they just go home. Either way, after the announcement, the hotel now has 1 empty room and an infinity of occupied rooms.
I understand the claim that any number that infinitely repeats a finite sequence after the decimal point is a rational number. I know it's a basic and conventional idea. What I'm saying is that this claim rests on the notion of limits. Without limits, I don't think you can even prove that 0.9-0.9=0? And I'm arguing that limits describe unending journey's down the Stern-Brocot tree (which can be used as the framework for arithmetic), not destinations on the Stern-Brocot tree.
That's a big difference. In one case a vacant room is magically created and everyone is happy while in the other case there is no magic, the room is vacant room because someone is standing in the hallway. Not being able to distinguish between these two cases means that the hotel manager's number system is broken. No matter what he announces he has no idea whether his instructions are being followed as he intended. How does your comment resolve the issue?
It's no difference at all. For the number system (i.e., the hotel and it's occupants), the resulting states are identical (room 1 is empty and every other room is occupied). There can be no hallway - that would be another room. When a person leaves room 1 without entering room 2, they have effectively left the hotel. As far as the hotel is concerned, they have ceased to exist.
But the whole point of Hilbert's hotel is that it can take in more guests. If it kicks guests out as it takes new guests in it's not actually able to hold more.
The problem is that you're focused on what happens to the previous occupant of room 1. Whereas your compassion is commendable, that's not how subtraction works. All that matters is the resulting state of the hotel.
Let's say you're holding 3 cookies and I take one from you. Does it matter to you whether I give the cookie to someone else, eat the cookie, or throw it on the floor? All you care about is that you now have 2 cookies.
Similarly, all the hotel cares about is that room 1 is now empty and available. Where the previous occupant went doesn't matter.
Hilbert's Hotel is a paradox because, in spite of being completely full, it can continue to make room for more guests without dislodging any of the current guests. If it made room for more guests by kicking out existing guests the magic is lost and nobody would care about the paradox. It wouldn't be a paradox.
Quoting Deus
What I said is true.
Quoting TonesInDeepFreeze
Yes, and it was axiomatized by set theory. What I said is true: From the axioms of set theory, we derive the theorems of calculus.
Quoting TonesInDeepFreeze
That is a sweeping and ignorant statement.
It was a steep you mentioned in defense of your argument. My criticism of that step is correct. And you presented your program to me also in defense of your argument, so I pointed out your program doesn't help your argument.
Wrong. We prove that any decimal expansion that has an infinitely repeating part represents a rational number. We may also conclude that 0.9[...] is rational because it is 1, but we don't have to do that just to prove that it is a rational number.
Quoting keystone
Yes, every denumerable binary sequence corresponds to a real number in [0 1]. And in a story tale, every binary denumerable binary sequence "codes" whether there is nor is not a guest in the room at that position in the sequence. No one disputes that.
Quoting keystone
Please provide a basis or source for that claim. I would have to refresh my memory; I don't recall whether the proof of "every decimal expansion that has an infinitely repeating part represents a rational number" must use limits. I am highly skeptical though of your claim that it does. Actually, though, since 'limit' is itself a defined notion, any proof that uses the notion could skip the notion.
Quoting keystone
That one is a whopper of ignorance.
(1) We prove the theorem Ax(x is a real number -> x-x = 0) without having to invoke the notion of limits. We do it right from the field axioms (and the definition of subtraction), which are themselves theorems of set theory per any of the constructions of the real number system.
Just considering the inverse law of addition is as basic as high school algebra. But here you flaunt your ignorance of even that. I never get over being amazed at the hubris of people who think they are showing flaws in mathematics that they know virtually nothing about.
(2) Since 'limit' is itself a defined term, no proof requires invoking it.
It never holds more than denumerably many guests.