The problem with infinity.

my_wan

It was Georg Cantor that demonstrated that not only was actual infinities perfectly logical but that it necessarily entailed orders of infinity, called aleph numbers [itex]\aleph[/itex] or cardinality (a powerset). Cardinality is basically the size of an infinite set. This was controversial in Cantor's day since the only infinity acceptable prior to that was unbounded sets, or potential infinities.

You can look up his work, including his diagonal argument, for a more quantitative treatment. I'll just give a more intuitive description. If you have a finite interval, i.e., distance between two points, then logically you can divide it into an infinite set of infinitesimal points. To answer the above question one way, it cannot be said that the infinite set of points between 0 and 1 contain all possible numbers. There is not only more than one infinite set, there are an infinite set of infinite sets. There is no flaw in Hilbert's Hotel.

There is also countably infinite sets and uncountable sets. There is also open sets and closed sets, which differ only in whether they contain the infinitesimal boundary points or not. There are also dense sets, which play a role in many quantum foundation arguments, including EPR and the scalability of quantum computing.

The one thing you cannot do is make a priori generalized statements, like infinity must contain the entirety of the whole Universe to be infinite. Any finite subset of the Universe also contains an infinite set of infinitesimal points. Neither can you, for the same reason, make the claim that two infinite sets must be the same size. You must restrict your statements about infinity to those statements that can be mathematically demonstrated to be consistent, and avoid the intuitively implied and inconsistent properties Zeno's arguments depended on.

bahamagreen

I'm not thinking that the infinite set of points between 0 and 1 contains all possible numbers, only that it contains all possible numbers between 0 and 1. It seems to me by definition, the set of points between 0 and 1 must include every point between 0 and 1. Are you suggesting otherwise?

I'm thinking that any arbitrary number I specify between 0 and 1 must already be included in the set of points between 0 and 1; so I don't see how any possible number between 0 and 1 is not already a member of the set of points between 0 and 1.

The "new guest" coming to Hilbert's Hotel's is like a point between 0 and 1 that is not a member of the set of points between 0 and 1... I see this as a flaw in the premise.

If the 0 to 1 range is problematic, we can do the same with the set of natural numbers... I'm thinking that the set of natural numbers must include any and all arbitrary natural numbers that I may specify... this seems clear by definition.
If each occupied room is mapped to a natural number, an infinite number of rooms means all the natural numbers are mapped, as are their corresponding guests... the "new guest" would need to represent an unmapped natural number, but there are none, by definition.

Maybe I'm missing something...?

Whovian

How does it remotely resemble this?

EDIT: Upon closer inspection, the paradox is simply mapping natural numbers to new natural numbers. The set ##\mathbb{Z}\cup\left[1,\infty\right)## has the same cardinality as the set ##\mathbb{Z}\cup\left[2,\infty\right)##. This should solve the paradox quite easily.

audioloop

or finite without boundaries, like a torus.

my_wan

In the original post I addressed it was implied that maybe if there was an infinite number of hotel rooms, then these rooms being occupied implied infinite guest such that there could be no new guest. If there are an infinite set of point between the two points, [0,1], and each of these correspond to a hotel room occupied by a point, then the original suggestion to work around the hotel hotel paradox implies that this infinity of points, [0,1], contains all points that might occupy the infinity of hotel rooms. Hence I made the suggestion in order to provide proof by contradiction that the hotel paradox was not flawed.

If the points between 0 and 1 are an infinite set of occupied hotel rooms, and yet "new guest" are still available from members that are not member of the set of points between 0 and 1, why is this a special case? The original suggestion was that an infinite number of guest implied no more guest exist, but here you add a special case to say there are more guest available from sets other that [0,1].


Precisely. The infinity problem is just as big in the interval [0,1] as it is in the interval [0,∞].

Only problem is that I can pull new guest from the infinite set of real number which you didn't included here. Note that the numbers are merely name tags on the guest, and it make no difference which ones you label with which numbers. I can relabel an infinite number of guest labeled with even numbers with odd numbers, and visa versa, and the count remains the same. I can also relabel all natural numbers as real numbers simply by multiplying their name tags with an irrational number and assigning them that number. It changes nothing about the total number of guest.

Endervhar

Cantor showed that all these infinities existed, but we should not lose sight of the fact that they are mathematical infinities. Mathematically, what does it mean to say that something exists? If a mathematician can write down a set of non-contradictory axioms, and set down rules for deducing mathematically true statements from them, then those statements can be said to ‘exist’. This existence requires only logical self-consistency. Physical existence is completely unnecessary.

If there can be a profound difference between physical and mathematical “existence” then it seems reasonable to identify a similar difference between physical and mathematical “truth”. Cantor’s infinities were all mathematical infinities, as are the rooms and guests in Hilbert’s Hotel. They may bear no relation to any possible physical infinity, which would include an infinite universe.

The actual, ancient, fear of infinity was not removed; it was just that Cantor provided the world with a “label” that could be attached to infinity, which reads: “this is a mathematical infinity – it doesn’t bite”.

bahamagreen

my_wan,
Your responses support my thinking that the Hilbert Hotel premise is flawed.

A "complete" mapping to the natural numbers would be to include all of them by taking the natural numbers in order... 1,2,3... any other mapping scheme like 2,4,6... is an obvious mechanism to skip some numbers, yet there would be objection to a scheme that skipped points between 0 and 1.
Claiming that the new guest could be from the set of real numbers when the set of guests is represented by the natural numbers misses the whole point. If the points between 0 and 1 represent the rooms, it is the guests that are mapped to the natural numbers. The paradox is based on the assumption that all guests, including any possible new guests, are all of the same class of thing - natural numbers, and violating that by suggesting a new guest could be a real number is like solving the question of the origin of the new guest by finding a chair and checking that chair into the hotel as a new guest... no, the new guest has to be a person like all the other guests.

my_wan

It doesn't miss the point any more than saying that the set of all hotel customers must consist of all possible hotel customers, and that is the only way you can claim there is nobody remaining to request a room in the hotel.

If the real numbers represent the set of all present hotel customers, and the natural numbers represent the set of all people that might request a room, then there are an infinity of people that may request a room even after an infinite number of people have already filled the hotel. To assume otherwise is effectively an attempt to impose a boundary condition on an unbounded variable.

my_wan

Very true, and so far the justification for "actual infinities" is fairly slim. However, attempting to avoid them has a number of problems. Avoiding actual infinities is just as problematic and paradoxical as accepting them. Modern mathematics didn't select the axioms simply to avoid contradictions with infinities, they where selected to avoid mathematical contradiction as a result of attempting to avoid them.

In the physical sciences the scalability of quantum computers actually depends on these mathematical properties associated with mathematical infinities. This comes from the fact that a Hilbert space must be "complete", i.e., is a complete metric space. The very thing that allows the calculus of limits, and from which we derive our mathematical justifications for infinities.

ObsessiveMathsFreak

It depends on what you mean by infinite. You really have to think about this one.

bahamagreen

The set of natural numbers must include all possible natural numbers.
The set of points between 0 and 1 must include all possible points between o and 1.
The set of all hotel customers must comprise all possible hotel customers.

You are defining:
The reals is the set of present hotel customers.
The naturals is the set of people that might request a room (not presently hotel customers).

I'm suggesting that is a flaw because you are defining some persons as both a non-customer and a customer - because the naturals and reals share some members in common (all naturals are members of the reals, some reals are members of the naturals).

But maybe I'm still missing something?

my_wan

This can't be justified. The set of all actual hotel customers anywhere in the world does not comprise all possible hotel customers. If that was necessarily true then in order for hotels to have any customers they must have every person on earth as a customer. Conversely, by this logic, since I am not a customer, either hotels have no customers or I am not a potential customer. Neither of which is true.

Endervhar

As ObsessiveMathsFreak pointed out, this "depends on what you mean by infinite." In terms of mathematical infinities your assertion is undoubtedly true, but it is a mathematical "truth" and has no significance in reality. You cannot have an infinite number of rooms, an infinite number of people, or an infinite number of anything.

Earlier, someone suggested that infinite and boundless might be synonymous; this is not so. Anything that is infinite is boundless, but not everything that is boundless is infinite.

Endervhar

Sorry, Nikkoo, I missed your quote when I was looking for it. You will probably want to take issue with my previous post. :)

my_wan

I have already gone over how it is relevant to the physical sciences, by way of Hilbert space. You can call Hilbert space a mathematical fiction, or slide rule of sorts, used to calculate. But a rejection of mathematically defined infinities has very real empirical consequences. The inability to scale quantum computers being a major one. Others involves issues surrounding Bell's theorem, and other no-go theorems.

You cannot escape the issues mathematicians have worked around with a "mathematical fiction" clause. Finite mathematics is inconsistent without mathematical infinities. How do you propose the reinstate consistency if you reimpose a finiteness condition on the physical world? You can't simply close your eyes and pretend it has no physical consequences, whether those consequences ultimately justify actual infinities or not.

Nessdude14

That's an absurd assertion. The number line is infinite, but we can still use integers to measure and divide segments of the line.

bahamagreen

How about addressing my previous post #28 first?