The problem with infinity.

my_wan

Yes, that is why I specified multiplying by an irrational, which is a subset of the real numbers but not a subset of the natural numbers. Hence the symmetry is complete.

Recap:
Set of all people (customers and potential customer) = real numbers
Set of all actual customers = natural numbers
Subset of non-customers (potential customers) = irrational numbers

Given that the set of all potential customers is larger than the set of all actual customers, whether irrationals are contained in the set of reals or not, there remains more potential customers than actual customer. I specified an irrational for the explicit purpose of of avoid a clash with your naturals (set of actual customers), in spite of the fact that there are more reals than naturals, which technically mooted the rebuttal anyway.

my_wan

bahamagreen,
I can sympathize with your difficulty on the hotel paradox. I have tried thinking through possible ways of getting around it. All of which involve refining definitions more than the paradox makes explicit. I'll try to construct a version of your argument that is harder to deconstruct. Though I will not offer any proof either. Neither does it reject infinities.

For instance, if you compare the statements:
(1) Hilbert's hotel contains an infinite number of rooms in which each room contains an occupant.
(2) Hilbert's hotel contains an infinite number of rooms and occupied by an infinite number of guest, for which the cardinal numbers are equal.

The hotel paradox essentially assumes these statements are equivalent. I suspect that this is not fully justified. There is only one countably infinite cardinal, [itex]\aleph_0[/itex], but there are uncountably many countably infinite ordinals ω. By definition in statement (1) we have assigned a one to one correspondence between the number of occupants and the number of rooms. Thus the one to one correspondence is in reference to ordinals rather than cardinals.

Now the equivalence of the above 2 statements is predicated on the fact that ω + 1 = ω, i.e., addition and multiplication are not commutative. Seems straightforward enough, just as 0*1=0 and 1 + 0 = 1. However, if we look to calculus, 0 may not equal 0, but rather an infinitesimal ΔL, the inverse of an infinity. In calculus we must make use of these limits specifically to avoid these self same ordinal properties we associate with 0, and inversely infinity. If calculus requires us to avoid this property with respect to zero, why is 1/ΔL special? ΔL simply has the equivalence class of 0, wrt a finite interval.

This wouldn't change much mathematically in operational terms, but would allow us to make a distinction between statements (1) and (2). In both ΔL and 1/ΔL the only thing that changes is the ordinal, not the cardinal. This would dictate that if the ordinal by definition has a one to one correspondence then there simply is no room to add another guest, though the cardinal remains the same up to [itex]\aleph_1[/itex].

We can also still accommodate more guest, when ω_1 = ω_2, under the condition that switching rooms requires some finite time interval, or a time interval with a cardinal number less than the cardinal number of guest.

Can anybody destruct that argument? It would be interesting to try and prove also.

Endervhar

I am not trying to impose a finiteness condition on the physical world. In fact I would reason that the cosmos (= everything that exists) must be infinite, otherwise we would not be here. The Universe may be finite; I believe there are very good arguments that it is; but this is simply the result of our limited perception. Here I am not talking about the limits of our equipment, but the fact that we observe everything within the restrictions of our 3+1 dimensions, which must not match the dimensionality of the infinite cosmos. Think of a spider walking through Flatland.

KUMAR SIDHANT

i agreed that universe is expanding regularly.. thats why we cant get the actual shape of it dude

InfiniteNiran

Something happens before something and something happens after something. It means there is always beginning for the beginning and there is always beginning after ending according to Thermodynamics law. Universe, Universes, Multiverses, Infiniverses and etc are incomprehensibly Infinite ∞

bahamagreen

It's been a little while, but I think I know what my problem is with Hilbert's Hotel.

The problem posed in the story is that the infinity of rooms are each already occupied. So the question is how to assign a room to the new guest...

I'm seeing an equivalence between the new guest and his new room. In looking at how both the guest and the room might be potential members of their infinite sets, the additional guest is allowed to exist and show up, but the additional room is not allowed to exist and thereby causes the assignment problem for the hotel manager...

Why is it that in spite of an infinite number of guests already assigned to rooms, another guest is allowed to exist, yet of the infinite number of rooms, another one is not allowed to exist and be found?

Another way to look a this is to break the thing into two independent questions:

1] Given an infinite hotel of rooms, are there any additional ones out there? Hilbert says, "No"...
2] Given an infinite world of guests, are there any additional ones out there? Hilbert says, "Yes".

From that difference he presents the paradox of solving the match up of guests to rooms... but why the two answers to the same kind of question? That is the premise flaw I see here...
Unless I'm still missing something.

mfb

You can look at a different setup where an additional guest shows up and an additional room is built, but that is trivial to solve.

Hilbert does not say "there is". The question is "imagine that, ... , how can we solve it?".


By the way: To make room for the additional guest, an infinite number of guests have to move. This is certainly annoying for them. And "a bit annoying" for an infinite amount of guests is worse than "very annoying" (sleeping in the corridor) for one guest ;).

bahamagreen

"The question is "imagine that, ... , how can we solve it?"."

If that is the case, then it is solved by noticing that the premise is based on a clear logical inconsistency.


If you imagine a hotel with infinite rooms, you have to apply the same logic to imagining an infinite population of guests... they are equivalent and need to be treated identically when considering the existence of an additional element.

The point of the premise is that two different conditions are being applied to two logically identical objects, one condition allows no new elements and the other does allow new elements.

Basically, the premise includes a guest without a room showing up, then more guests, then bus loads of guests, etc. If there are infinite guests already in the rooms of the hotel, and more guests are allowed to appear, then the same applies to the rooms; there are an infinite number of rooms, but more can be found.
To say no more rooms can be found is the same as saying no additional guests can appear.

"Infinite" may or may not entail "all", but either way needs to be applied to both the rooms and the guests...
If infinite means "all", then the infinity of guests in rooms already means no additional guest can show up. If infinite does not mean "all", then there is at least one additional unoccupied room in the hotel.

mfb

@bahamagreen: That does not make sense.

Imagine I have 3 bananas and give them to 3 monkeys. Each monkey gets a banana.
Imagine I have 4 bananas and give them to 3 monkeys. You can imagine that, right? Even after I added a banana.

It is an imaginary situation, I can use any numbers I like.
I can distribute infinite bananas on 3 monkeys - at least one monkey has to get an infinite amount of bananas, so what?
I can distribute infinite bananas on an infinitely many monkeys. And then I can take another banana and give it to monkeys.

Dead Boss

It doesn't matter whether new rooms can be found or not, because you can comfortably fit the new guests in the rooms that are already occupied.

bahamagreen

You're not getting the point.
Forget about the methods of putting extra guests into the rooms.

Look at the facts of the problem:

Hotel has
Infinity of guests
Infinity of rooms

If you allow that an additional guest (without a room) can exist,
you must also allow that an additional room (without a guest) can exist.
These two things are the logically identical, therefore the premise that the new guest has no room is false.
If a hotel with infinity of rooms does not have an empty room available, then likewise, with an infinity of guests with rooms, there will not be possible a new guest without a room.

Look at it this way; what if the original paradox had been reversed?
The hotel has an infinity of rooms and guests...
Then one day a new empty room is discovered, but the manager needs to report full occupancy to be paid his bonus.
So he shuffles the guests through the rooms (including the new one).
The simple solution is for anther guest to appear (like in the original).

See, the discovery of a new guest is just like the discovery of a new room.

dyb

If the universe were infinite and existed for an infinite amount of time and the universe weren't expanding, and the universe had the same kind of distribution of galaxies everywhere, then night would be bright as day because no matter what direction you look there would be light coming from that direction.

But, due to red-shift, far away galaxies may no longer appear in the visible spectrum (but you could measure it). And if the Universe expanded fast enough outer parts could become "disconnected". So it's a little harder to figure out.

Endervhar

Bahamageen, it’s refreshing to find someone who thinks along the same lines as I do about infinity.

I believe the main problem is that since Cantor “tamed” infinity there have been two distinct perspectives; the mathematical infinity and the real infinity. Much confusion arises because there is such a profound difference between the two, yet we can discuss infinity with one person using one form and the other person using the other form.

Mathematical infinities are simply mathematical concepts that are of value in some calculations. There exists an infinite number of mathematical infinities which can be of different sizes.

Your arguments apply to real infinity. There can be only one, and it must include everything. If you try to introduce numbers to such an infinity your calculations lead to nonsensical answers.

. Take for example the (UK) national Lottery. In an infinite universe, an infinite lottery becomes possible, and therefore inevitable, not only that, it must occur an infinite number of times. So, what would this infinite lottery be like? There would be an infinite number of people taking part, the staked money would be infinite, therefore, the jackpot (being a percentage of the stake) would also be infinite, the jackpot winners (being a percentage of the infinite number of people taking part) would be infinite, as would the number of losers. We can see from this that an infinite number of people would win an infinite share of an infinite amount of money, but, paradoxically, the same infinite number of people would not be winners at all.

Mathematicians can find their way, logically, through Hilbert’s Hotel, followed by infinite guests grumpily changing rooms; but in the real world it calls to mind the well known debate about angels and pins.

Dead Boss

You're not getting the point that it's not important whether there is a room for the guest or not: you don't need it even if there is. You can fit the new guest into one of the old rooms and not use any new room. The premise that there is no room for the guest is not a premise of the mathematics of the paradox but only of the "story" behind it. The simple mathematical idea is wrapped in a textual anecdote and that anecdote requires the "no room" premise for the problem of fitting the new guest to exist. You can just as well state that the hotel manager is a freak who loves to move guests around. Nothing changes the fact that you can find a bijective function from integers to naturals (for example) which is all this "problem" is about.

bahamagreen

I understand it's just a story... but the "story" has a flaw, an inconsistent logical treatment of the rooms vs the guests.

It makes no sense to allow for a "new" guest to be found and them maintain that a "new" room can't be found.

I understand that IF you take the first statement as a given, THEN yes, the story acts fine as a puzzle for how to place the new guest in a room, in spite of all the infinite rooms already being occupied by the infinite guests.

The problem with this story is that anyone who thinks about the origin of the new guest must logically conclude that the first statement is inconsistent... that statement being that no new rooms can exist.

I'm pointing out a separation between two things:

The story as presented (which may be fine for setting up how to solve a problem with infinities)
and
Looking at the problem on its face and realizing that it is a false problem.

The reason I think this is important is because if the question (story) is allowed to contain illogical inconsistencies, what rules then prevent the answer from also using logical inconsistencies?
If the set up for the story is "wrong", then how is any answer "right"?
Or how can any answer that just magically and illogically "solves" the problem be legitimately denied?

I expect the rules that apply to evaluating the answer to be the same rules that must apply to the construction of the question.
Hilbert's Hotel does not meet that expectation; the problem needs to be logically repaired before being asked, and after the repair it is no longer a problem, no paradox.

For learning purposes, maybe the problem should be altered so as not to raise the question about where the new guest came from by having the hotel originally occupied by an infinity of women, and the new guest is a man (a lucky man!), then do the subsequent movements of occupants to insure one person per room is enforced... (not so lucky man).

dyb

Perhaps an analogy you will understand:

Consider the number 10/9. It is 1.1111111... with an infinite number of 1's after the decimal point. Now you may argue that 1/9 is the 10th of 10/9, so the decimal digits get shifted by one to the right and therefore there must be one additional 1 after the decimal point. So, you could argue that 10/9-1/9 would need to be slightly below 1. But it's not. It's exactly 1 because 10/9-1/9 = 9/9 = 1.

(This also follows from the definition of decimal using limits)

ImaLooser

Nah. The story tells you that all of the rooms are full.
There is no room without a guest.