# Can hypothetical questions with impossible criteria really be on-topic?

I have not been on this site very much, I must admit, but I stumbled on the question Can I have visited all rooms of Hilbert's infinite hotel? and before I put forward why I am posting about this question, I would like to make a few points.

## Mathematics is all about possibilities

In a way, mathematics is scientific, because it is a study of what is possible, even in theory. You cannot write anything mathematical about something which cannot possibly occur.

## There can only be one answer to the question I highlighted and that is No. It would be impossible, as I answered today

It just stands to reason that when you have an infinite number of rooms, you cannot possibly stay in every room by changing rooms each day.

The number of days you have stayed will always be finite. Even if you were able to stay for eternity, no matter when you assess the number of rooms there will always still be an infinite number of rooms still to stay in.

## This question is a hypothetical question and it seems there can be no possible answer which would satisfy the OP.

The reason why I say this is because in the comments, the OP states that

For this thought experiment it is crucial to say that I have been staying at the hotel for an infinite amount of days. That seems weird to some people but I don't really wanna share my thoughts behind this question, it is simply important to assume that I have already been staying at this hotel for an infinite amount of days.

## How is this question able to be on-topic?

Your length of stay to date will always be finite even if hypothetically you could stay for eternity. You cannot possibly stay in every room.

It would be pointless to say

Ah, but what if it were possible to have stayed an infinite number of days?

because the theory just isn't possible.

What have I missed here?

## OK, to close a question for this reason you may need to first prove the impossibility, but...

Should we not be asking for mathematical foundation for the hypothetical question in the question when asking hypothetical questions?

• -1 for mathematics is the study of what is possible. I enjoy mathematics because it's the study of what is impossible. – YuiTo Cheng Apr 14 at 7:49
• @YuiToCheng - if so, can you please give me example web links to mathematics of the impossible so I can learn this? – Chris Rogers Apr 14 at 7:53
• It seems like you are a finitist, right? – YuiTo Cheng Apr 14 at 8:05
• @YuiToCheng - If you wish to pigeonhole me, I would say I am an open-minded. I have never had the cause to work with mathematics involving infinite numbers, but I understand and accept the concept of infinite numbers. The thing is, a number can either be finite or infinite. It absolutely cannot be both, which to my mind, Hilbert is trying to put across with his "Grand Hotel paradox" (a fully occupied hotel with infinitely many rooms may still accommodate additional guests). There is no paradox. It could never be full – Chris Rogers Apr 14 at 8:52
• Why couldn't Hilbert's Hotel be full? – YuiTo Cheng Apr 14 at 9:06
• You're confusing this site with Physics. – Asaf Karagila Apr 14 at 9:18
• "Alice laughed: "There's no use trying," she said; "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." [The author, please note, was a mathematician.] – Gerry Myerson Apr 14 at 12:22
• "Can hypothetical questions with impossible criteria really be on-topic?" Well, on-topic is a technical term here, defined to mean, "not off-topic". And off-topic is defined to mean, "voted off-topic by five users". So a question, hypothetical, impossible, or otherwise, is on-topic until five users vote to the contrary. If you think the question is off-topic, fine: vote to close it (if you have enough points to do that). It's hardly worth getting so worked up about it. – Gerry Myerson Apr 14 at 12:30

If you think a question should be closed for being off-topic, that's what flags are for. You can also go to chat-rooms like CRUDE and make your case there. Without wanting to sound rude, this question comes across a little like a rant.

However: there was a time when negative numbers were 'impossible' in maths. And a time when the square-roots of negative numbers were 'impossible' in maths. And a time when 'infinity' was just a vague concept with no real understanding behind it. Had people at those points said "well, it's impossible, so let's not talk about it" we might not be where we are today. [As a side-note it's probably worth saying that there were plenty of mathematicians who disagreed strongly with Cantor's thoughts on infinity for a long time, so it's certainly not the case that this never happens. Kuhn's notion of a paradigm shift is worth thinking about here.]

The claim you make: no one can visit an infinite number of rooms so the question doesn't make sense, accepts that there can be an infinite number of rooms in the first place, which would appear to make no sense to you either -- if you allow an infinite number of rooms then you should allow an infinite amount of time in which to stay in them all (not that you need to: if you visit each room using half the time you took to visit the previous room you get there in finite time; this might not fit with most people's notion of 'staying' in a room, but it's still there). So really you should be objecting to all of the notions in the question. However, you go on to say that this is a hypothetical question -- well, pretty much everything at the bottom of mathematics is hypothetical: points are idealised objects that have no dimensions, so they're hypothetical and since lines join them (being made up of... you guessed it, an infinite collection of points) they're hypothetical and suddenly all of geometry is objectionable.

Hypothetical questions aren't a problem. Seemingly silly questions aren't a problem (parallel lines never converge on flat surfaces, and for most of us, that's everywhere we look yet 'what if they do converge' leads to two new types of geometry, both of which have applications). What is a problem is an unwillingness to explore and understand. Maybe, after enough thought and work we decide that the question is somehow unsound (Qiaochu Yuan says that on the question you link to) and we can correct it. But we do that from the basis of a solid understanding (ideally).

• You seem to say that it is possible to square root a minus number. That is curious. As someone who studied electronic engineering, I understand the concept of $\sqrt{-1} = i$ (or in the case of electronic engineering $i = j$ to avoid confusion in electronics equations), but can you point me in the direction of some explanatory text to tell me how -1 can actually be square rooted? – Chris Rogers Apr 14 at 9:04
• You're going to have to set out your beliefs first. If you want me to show you some other real number that is the square root of minus one, obviously that doesn't exist. The square root of minus one is $i$ (or $j$) just like three is $3$. Now, if you want to say three exists because you can set out three apples, but the square root of minus doesn't because you can't represent that in apples, you need to be on the Philosophy stack exchange :) – postmortes Apr 14 at 9:09

If there exists "an educatory component" then it might be instructive for the OP (and for the advantage of future readers of MSE) to be explained how the question has improper premises. (I'be been teacher in statistics and had often to handle with making premises and their associate problems explicite and visible to the students)

So, if you can detect the possibility for such "educatory component" then please answer. If it is obviously (when is something obvious?) a trolling question you might vote to close and comment. (Or even flag for moderator's attention) The concept of MSE-community mostly helps to find consent with others to improve or correct the "obvious"-feeling.

Thanks

# Thanks again

Upps, well, did I say thanks already?

# Ok, to not miss it: Thanks again!

• Is it not helpful to have each point headed in order to jump to the relevant point when re-reading in order to respond to each point in the question? I think th(ese) point(s) in your answer are a bit rude in my mind – Chris Rogers Apr 14 at 9:15
• @Chris - well I think everything has a "metric"/a measure. When is it too much, when too little? If there is 70 lines then 2 or three bullet lines might be ok, in my taste. But in a mere 20 or 30 liner 4 or more bullet lines? Well - my contribution is only a plea for reconsidering the general amount of enhancing. – Gottfried Helms Apr 14 at 9:19
• This post seems to me to be needlessly unfriendly. – Frank Hubeny Apr 14 at 12:04
• @Chris - don't want to make this an endless stream... But just for instance, what I mean with structuring a posting but not overusing bullets, bigfonts, colours etc. I think "parsimonious" is the correct english word? (I'm not native english speaker) See that structured answer of mine: math.stackexchange.com/a/3186011/1714 but also the question itself has already structure, but I intended that there's no sideeffect of distraction attention from the (quiet) reading. – Gottfried Helms Apr 14 at 15:49

Let me comment at least briefly on the topic of the linked question - Hilbert's hotel. I do not think that it falls under "hypothetical question with impossible criteria".

Hilbert's hotel is an analogy which helps to illustrate some properties of infinite sets. Typically the stuff discussing Hilbert's hotel can be rephrased using infinite countable sets and some functions between them and operations with them. So questions about Hilbert's hotel can be (almost always) rephrased in a rigorous way using sets such as $$\mathbb N$$, $$\mathbb Z$$, $$\mathbb N\times\mathbb N$$, etc. (Depending on the situation.)

Formulating the question as a question about infinite hotel has the advantage that it makes the topic more accessible to the people who do not have much formal knowledge about infinite sets, cardinalities, bijections, injections. And even in the courses where students are supposed to learn these topics, Hilbert's hotel might be useful to help them visualize things - and also to enliven the lesson a bit.

So questions about Hilbert's hotel do not have a hidden assumption that there actually is a hotel like this and that the guests would be physically able to move between infinitely many rooms. (Similarly as you don't see animal right activists to get worried about experiments with Schrödinger's cat.)

While the question linked above might be considered not sufficiently rigorous and it could possibly put on a more rigorous basis, this analogy is known well enough that people with enough experience in set theory know that questions about Hilbert's hotel are actually questions about infinite sets. (Of course, depending on the question it might be possible that there are several possible interpretations.)