Hilbert's paradox of the Grand Hotel
en.wikipedia.org
en.wikipedia.org
Technically, though, there aren't any paradoxes here. It's for the purpose of avoiding paradox that we're willing to talk about infinities in the first place.
The real reason why it is a paradox doesn't have much to do with notion of infinity per se, but with problem of infinite decision aka Supertask.
Which reminds me of an article on physics of supertasks that among other things somewhat outlines how classical theory is more paradoxical before than after quantization http://th-www.if.uj.edu.pl/acta/vol36/pdf/v36p2887.pdf (43pp but easy reading).
Accommodating infitine number of newly arrived guests thus means having infinite number of poor "homeless" guys in the lobby.
So? So... Grand Hotel can accomodate exactly infinite number of guests, but not more than that. If it's full, it's full.
Let alone the fact that infinity doesn't exist because we can't observe it.
I always have the sneaking suspicion that these things only feel paradoxical because we have a better intuitive understanding of the "First" room than the "Last" room, and yet we also understand the idea of the "Next" room. In every case, we have a chain of "Nexts" taking us from the concrete "First" to the weirdness-factory "Last".
Let alone the fact that infinity doesn't exist because we can't observe it.
Does someone standing silently behind you cease to exist until you turn around?
No, because once the first client moves, the first room is free for the new guest, even though the whole process of moving hasn't finihsed yet, and in fact will never be finished. You will end up having infinite "waves" of moving clients.
assume that no guests have possession, and that every room has a door to the previous and next rooms
My point is, there are intermediate states when you can't strictly say the hotel is full and that every occupant has a room, i.e. one-to-one relation. In your scenario, if you start from the first room, you will always have one room with two occupants, which is against the rules, or in my scenario there will always be one homeless dude.
I understand this is an abstract situation anyway, which can not be experimentally reproduced (for one, it would take infinite time to build this hotel and we'll never see it finished), which leads to the next point:
Does someone standing silently behind you cease to exist until you turn around?
You can never be sure about anything unless you observe it using your own senses and possibly some equipment in the middle. (And it gets even worse in the quantum world, but let's forget that for a moment.) So strictly speaking, when you don't see the person behind you, you can only make a prediction/hypothesis that if you turn around you will see that person again. It's based on some laws of physics suggesting that a person can't disappear, say, in one nanosecond. But hypothesis is just a hypothesis, and it's not observation of reaality, e.g. a fact.
Back to infinities. In the physical world all kinds of infinite values are defined through either infinite distance or infinite time, including both infitely big or small, i.e. ∞ and 1/∞. Space and time are closely related though, because in order to measure a distance you need "some" time, and obviously, to measure or observe infinite distance you need infinite time. Which is not possible for us, humans and therefore infinity is just a hypothesis, and I'm afraid will always be. (Always? You mean, for infinitely long time? :)
Curiously, for some reason I started thinking about infinities this morning and here you go, a post at YCNews.
There is a pretty good Overcoming Bias post on why this is wrong:
So he understands that when we lose sight of that photon, from that point on our knowledge about it becomes a hypothesis based on laws of physics. Unfortunately laws of physics are neither 100% accurate nor complete - there may be some laws not discovered and formulated yet and even those discovered never give you 100% accuracy in making predictions.
However, we are talking about infinities, which is a different kind of beast. My main thesis is that in order to measure infinite distance you need infinite time, which we simply don't have at our disposal, and thus infinity in any of its form can't be observed in principle.
Presumably, you are accessing this website using a some kind of personal computer and some communications infrastructure. The engineers who designed these systems all made frequent use of the notion of infinity, particularly in their use of limits and calculus, when these systems were designed.
I am sure that those engineers would love to hear about the equivalent (but necessarily much more complex), infinity-free versions of calculus and limit processes that you have obviously developed. Since infinity is the only simplifying assumption the average engineer is ever forced to make, I'm sure that as a group, they will be very eager to take your advice, even if it means they have to do twice as much work.
Take any serious book on modeling bus and routing infrastructure (and not vocational guide on using protocols designed by someone else) and you will see plenty of partial differential equations.
Edit: a theory that involves infinities.
Edit: I'd like to warn you though, that because I'm more or less familiar with the subject, I think it will be very difficult to find what I'm asking for.
Cool. Rather than posted challenges, how about educating us to why we are wrong. If you know this area well I'd be interested to learn more.
The engineers who designed these systems all made frequent use of the notion of infinity
and I'm yet to see the proof.
The engineers who designed these systems all made frequent use of the notion of infinity, particularly in their use of limits and calculus.
Do you disbelieve that engineers make frequent use of limits and calculus?
Do you disbelieve that notions of limit, derivative, and integral are contingent on the idea of infinity?
Why should anyone answer your questions while you continue to ignore mine?
Although in some narrow areas I've seen non-trivial calculus, but I barely remember anything that involves infinities and is significant for the solution.
Now you seem to be saying: "I have never, myself, seen the concept of infinity used to solve a problem in a way I deem significant. Therefore, infinity does not exist and is not useful to anybody."
If I said the same thing about the number 65536, would it force you to program only 8-bit processors for the rest of your life?
Based on some of your earlier statements, I'm going to guess that we're using different definitions of the word "infinity." I'd guess that to you, the word "infinity" means something like "the largest possible number, denoted ∞." I'd also guess that, in your claim about calculus above, you mean something like "I have never seen a calculus problem whose solution required you to write down an ∞." Am I right?
Since the MRI machine only samples the signal coming in it uses a DFT, but that doesn't negate the fact that the theory of Fourier Transforms involves the notion of infinity and that engineers used that body of mathematics to build a real machine.
Ideally, a theory based on which engineers built something, but not the other way around, i.e. when something exists and some theorists are trying to build a de facto model of it. Edit: a theory that involves infinities.
Spot the contradiction in this sentence. Even if I would like to argue with you on unedefined problem you have, I can't satisfy your requirements when they're contradictory.
I see PDE's in TCP papers, I don't know what's your problem.
Of course I expect you will dismiss that as too "astral" of an example, but ask Vint Cerf if he did not need any calculus. And then ask him (or Lloyd Wood) again about his new suite of protocols designed for guaranteed connectivity in spite of enormous latencies in orbital and interplanetary internet (with cisco routers with PDE-optimized buffer sizes orbiting already on several satellites).
Why didn't he just ask about discrete mathematics, then?
Asymptotic analysis of the computational complexity of algorithms.
There are, of course, things that are described by infinity: the energy required to accelerate a mass to the speed of light, the number of points at which a magnetic field can travel through a surface, etc.
The number two is a generalization of quantity. We know there may be two apples or two birds on the wire, and at one point we are taught quantity can be applied to virtually antyhing.
Now, what's infinity? Another abstraction of quantity, which has many definitions. However, every definition you take involves infinite time, e.g. you add 1 to a number ad infinitum or you accelerate a particle to the speed of light, and because reaching precise speed of light is not possible, you will be doing it infinitely.
So even "we use them to describe things we find in nature" - even that is not true. You can't find infinity in nature, neither anything that can be described with infinities.
Do you see where the problem might be?
(If this is some sort of attempt at a joke, it's not very good.)
Do you feel the same way about the complex numbers?
The integers (like the hotel rooms) are countably infinite. There are only a FINITE number of integers (or hotel rooms) between any two integers (rooms), like between rooms 1 and 100.
The set of real numbers is a larger set than the integers. Both are infinitely large, but the real numbers are a larger infinity. Between any two real numbers, (even 0 and 1), there is still an INFINITE number of real numbers in between.
The notion of a hierarchy of infinities initially seems paradoxical, but once you know how to distinguish countably infinite from uncountably infinite, you have the key concept.
One way to see that there are as many integers as there are rational numbers is just to find a way to count the rational numbers. There are a number of ways to do this. There is a very nice and explicit way to do enumerate the positive rationals that involves the prime factorization of the integers. Suppose first that p is a prime number. We identify p^k with the integer p^{2k}. For the rational number 1/p^k (p is still prime), we label it by the integer p^{2k-1}. For a rational number p/q in lowest terms, we take the prime factorization of p and the prime factorization of q, apply the above identification to each factor p_i ^k and then take the product. This gives a very explicit bijection between the positive rationals and the positive integers.with
My favorite way, though, is just to make a grid of pairs of integers (numerator, denominator), and then count them by spiraling outward on the grid. This yields duplicates, but that's ok.
Calkin and Wilf cleverly avoid duplications by arranging reals not on a lattice but on a (Stern-Borcot) tree. http://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree http://www.mathlesstraveled.com/?p=94
Or just enumerate the naturals in base 11. Denote 9+1 by "/". Interpret "/" as a negative sign when it's the first or last base-10 digit, otherwise as division. Pick an associativity.
I first got embarassed by not knowing of that direct way here: http://gowers.wordpress.com/2008/07/30/recognising-countable...
To discuss a necessary step in mathematical induction, namely the basis.