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. — PhilosophyRunner

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+...].
All the irrational numbers would appear at the bottom of this tree...but once again, there is no bottom of the tree. So when I think of pi, I think of an endless journey down this tree that begins with steps RRRLL. Rationals are destinations on the tree. Irrationals are journeys. I know LR and LR look similar but they're fundamentally different.

The reason why you think 0.9 is rational is because you believe it equals 1, which is indeed a rational number.

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. — Real Gone Cat

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

The reason why you think 0.9 is rational is because you believe it equals 1, which is indeed a rational number. — keystone

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 $\infty$ 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.

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. — PhilosophyRunner

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.

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. — Real Gone Cat

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.

[From] the axioms of set theory, we derive the theorems of calculus. — TonesInDeepFreeze

Ridiculous. — Deus

What I said is true.

Calculus was developed well before set theory came into the scene. — 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.

Also in the field of mathematics it’s nothing more than a minor development/distraction — TonesInDeepFreeze

That is a sweeping and ignorant statement.

Your criticism of my story was of an inconsequential intermediate step. And even now, you focusing on the program is secondary. — keystone

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.

The reason why you think 0.9[...] is rational is because you believe it equals 1 — keystone

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.

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 — 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.

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. — 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.

Without limits, I don't think you can even prove that 0.9[...]-0.9[...]=0? — 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.

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. — keystone

It never holds more than denumerably many guests.