Could a hotel with an infinite number of rooms be packed?

[Hurkyl]

Yes I am serious, there is no way to comprehend infinity because it is defined as such.

Well, maybe that the problem, then. You're interpreting the word "infinity" in some strange way that nobody else is using.

 

[Evolver]

This is what you said. What's the evidence that our brain can never think of a hotel with an infinite number of rooms? Conceptually, it's done all the time. The problem can be stated very specifically.

RE your post #48

'Paradox' like may words, has more than one definition. It can be an apparent contradiction which in fact may be true. Because the word is ambiguous, it's not the basis for a yes/no answer.

Take a finite number 'a' from a set with x members such that a<x; then a/x goes to zero at the limit as x goes to infinity.

Agreed, but there are also multiple definitions of an infinite set. It can be a countable infinite set or an uncountable infinite set. Some infinite sets are allowed to be greater or lesser than other infinite sets. The hotel paradox does not define any of these qualities of an infinite set and thus leave it to interpretation.

The brain cannot fathom an infinite set that is essentially endless. They may be able to use logic to deduce certain implications of the infinite set (like if it's infinite it can always have more rooms, etc.) But that doesn't really mean anything in the essence of truly understanding infinite. Because as I say, it's a placeholder for an unfathomable idea. For instance, to give an analogy from physics... when something becomes 'infinite' it is when the rules of physics break down. (Ex., singularity, infinite mass at light speed, etc.) Scientists don't know what happens in the singularity or even attempt to know, but to say it is infinite is a placeholder for that thought. In mathematics it is more conceptual in nature, but it does not mean it is understood in any classical sense.

 

[Evolver]

Well, maybe that the problem, then. You're interpreting the word "infinity" in some strange way that nobody else is using.

There are multiple definitions of an infinite set... countable, uncountable, greater or lesser, etc... I'm very well aware of the ways in which to interpret it.

 

[Hurkyl]

Agreed, but there are also multiple definitions of an infinite set.

AFAIK, there is pretty much one definition of "infinite set" -- not finite. (With "finite set" meaning, roughly, that its cardinality is a natural number)

What you subsequently describe is that there exists more than one set that is infinite, and that their cardinal numbers may be of different sizes.

The brain cannot fathom an infinite set that is essentially endless.

I am not limited by your lack of imagination. And besides, there are infinite ordered sets that do have ends. \omega + 1, for example.

(P.S. it doesn't make sense to talk about the "end" of a set, so I assume you were talking about ordered sets)

And besides, I'm pretty sure I have an easier time "fathoming" set of cardinality \aleph_0 than I would "fathoming" a set of cardinality 129848300199285771.

Because as I say, it's a placeholder for an unfathomable idea.

You may be right about whatever you mean by "infinite", but you are the only person talking about that -- the rest of us are content to talk about mathematics and set theory.

For instance, to give an analogy from physics... when something becomes 'infinite' it is when the rules of physics break down. (Ex., singularity, infinite mass at light speed, etc.)

The rules of physics only break down when something becomes infinite if they rules of physics say they do.


Note, for example, that classical mechanics has absolutely no problem with the fact that the number of points within a box has cardinality 2^{\aleph_0}...


Also note, by the way, that all this talk about infinite sets and cardinality and whatnot has nothing to do with the extended real numbers +\infty and -\infty that arise in calculus and physics.

Scientists don't know what happens in the singularity or even attempt to know,

You're talking about a black hole, right? FYI, in GR, the "singularity" is simply the edge of the universe.

 

[Evolver]

AFAIK, there is pretty much one definition of "infinite set" -- not finite. (With "finite set" meaning, roughly, that its cardinality is a natural number)

What you subsequently describe is that there exists more than one set that is infinite, and that their cardinal numbers may be of different sizes.

True infinite sets are not finite, but that does not mean all infinite sets are the same... Cantor's Diagonal Argument shows that. There are countable and uncountable infinite sets. Some are allowed to be greater than others. For instance integers vs. real numbers... those encompass two separate infinite set types with two different qualities.

I am not limited by your lack of imagination. And besides, there are infinite ordered sets that do have ends. \omega + 1, for example.

(P.S. it doesn't make sense to talk about the "end" of a set, so I assume you were talking about ordered sets)

And besides, I'm pretty sure I have an easier time "fathoming" set of cardinality \aleph_0 than I would "fathoming" a set of cardinality 129848300199285771.

Imagination has nothing to do with it... it's about having the capacity to comprehend something. A mouse couldn't comprehend calculus no matter how much it tried... would I then say the mouse has no imagination, or simply that it is not adapted to comprehend such things? A mouse would have equal difficulty comprehending advanced calculus as it would high school geometry.

I'm not arguing against humanity or set theory or anything. I'm just calling a spade a spade in the sense that the only way we can attain higher thought is through representative notions and models... and that is precisely what set theory is.

The rules of physics only break down when something becomes infinite if they rules of physics say they do.

But of course, if there were no rules there would be no ideas... set theory has rules too... Like the hotel paradox for example. In the hotel paradox, a hotel with infinite rooms filled by infinite guests can accommodate a new guest, or a bus load of guests, or a bus with infinite new guests, or infinite buses with infinite guests. Infinity modified by 1, or 20, or even infinity is still infinity. Yet there are infinites which can be greater or lesser than other infinites. These too, only exist because set theory says they do. Does that mean you can comprehend it any more or less than you can the laws of physics breaking down?

You're talking about a black hole, right? FYI, in GR, the "singularity" is simply the edge of the universe.

Singularity as in the point where physics (and mathematics) break down, whether that be the crushing gravity of a black hole, or the singularity at the beginning of the big bang. A mathematical singularity is no different... it is the point where a mathematical object is not defined.

 

[disregardthat]

Imagination has nothing to do with it... it's about having the capacity to comprehend something. A mouse couldn't comprehend calculus no matter how much it tried... would I then say the mouse has no imagination, or simply that it is not adapted to comprehend such things? A mouse would have equal difficulty comprehending advanced calculus as it would high school geometry.

Imagination may have something to do with it if certain aspects of infinity is in principle unimaginable.

 

[debra]

The infinity notions have practical problems that even the universe would meet. There must be a limit to the degree of precision in a system or process.

A curve for example must be discontinous at some level, otherwise the steps needed to produce a curve would need to be infinitely small (pixelated). The metric of space time needed to define such small steps would be unmanagabley large.

A computer with a mega trillion bit processor using half the universes energy would still fail to draw a perfect curve.

 

[Pythagorean]

Under what conditions would it be possible for the universe to even be able to define an infinite quantity that we could satisfactorily interpret?

This is exactly why OR rules it out. It's not very helpful to us if we can't access it and make use of it.

 

[Pinu7]

This would depend on what you mean by "infinite amount"
There are two types of infinity

1. Countable infinity("Aleph-null"): A set has the cardinality of countable infinity if a bijective map can be created with the set of natural numbers.

2. Uncountable infinity("continuum" or "c"): A set has the cardinality of infinity if a bijective map
exists between the set and the interval [0,1] on the real number line.
(See: Real Analysis-Kolmogorov)

Denote the set of rooms by R, and the set of guests by A. There are five possibilities(assuming R is infinite).
1. card(R)=Aleph-null, card(A)=/=Aleph-null this means it will not be packed.
2. card(R),card(A)=Aleph-null this will be packed.
3. card(R)=c, card(A) is a natural number there will be infinite(c) free rooms.
4. card(R)=c, card(A)=c this will be packed
5. card(R)=c, card(A)=Aleph-null It will be packed, and there will be unroomed
guests

 

[Tinyboss]

There are infinitely many distinct infinite cardinalities, not just two. Cantor's diagonalization argument assures that the power set of S has cardinality strictly greater than |S|. So |S| < |P(S)| < |P(P(S))| < ...

Edit: I realize that all infinite sets are either countable or uncountable, but the fact that there are distinct uncountable cardinalities means we can't reduce this to a finite number of cases.