The Hilbert Hotel
opinionator.blogs.nytimes.com
opinionator.blogs.nytimes.com
In fact, I'm coming to believe that the usual methods of teaching infinity are wrong (not the concepts, just the presentation).
People have this intuition that there can't be anything bigger than infinity. We reinforce that by showing that the rationals are countable, that Q^2 is countable, in fact Q^n is countable. Even the algebraics are countable. We create the intuition that anything infinite is countable.
Then we present 2^N and everything goes bizarre. Some people like that, but others decide that it's all meaningless. I think this method of presentation does the majority of people a disservice.
But not only is there the set-theoretic infinity, there is also the geometric infinity. That actually better matches people's intuition. Separating the ideas of the set theoretic and the geometric versions of infinity has, I'm finding, huge benefits when trying to help people understand what's going on.
Of course, there's also cardinal versus ordinal infinities as well. That's fun.
Expand on that please, so I can make sure I understand what you are writing here.
This is a brief reply - I don't have time for more detail now, I am intending to write this up, but it's about 50th on a very long "To Do" list.
People have this concept that infinity is kind of, well, "out there", as far as you can go. You can't go any further, it's all there is. People talk about parallel lines "meeting at infinity" and "there is no last point on the line" sort of thing.
And that's what their intuition is telling them. There's nothing beyond infinity.
Then when we talk about the cardinals we tend to reinforce that by showing that the odds, evens, squares and primes all have the same cardinality. That's a bit weird for people, but they're getting the idea that infinity is odd, but they can cope with odd.
Then we show that N^2 and Q are both countable, can be put in one-one correspondence with N. They're getting cool with that too. We continue to reinforce the idea that there's infinity, and every time we do something we get the same infinity.
But that's now what they expect. Infinity is kind of as far as you can go. We're matching the set-theoretic cardinals to their geometrical intuition of "out there."
No wonder they get confused, upset, and occasionally angry when we then introduce 2^N as being "bigger."
I'm finding that talking about geometry and the concept of infinity, then talking about sets and the concept of intinity as a different thing, despite the same name is helping people to create different models in their minds. These different models then help them not use the same intuition, and they don't get confused in the same way.
Then, as the final modification, I don't start by showing that loads and loads of things are all countable. I talk about "same size" as being 1-1 matching, and discuss the idea that there could be something "bigger than N."
I do that first, and then go hunting, pointing out from the beginning that historically people found this difficult.
Now they get excited when I show them that 2^N is uncountable. They've been primed to want to find it, and their intuition about "infinity" isn't challenged.
That's an incomplete summary - I hope it helps. Ask for more, or email me if it's not clear.
From that point of view, it would appear that your intuitions are not helping you. More, under some models of geometry it makes a lot of sense to say that parallel lines really do meet at infinity. It makes some theorems a lot easier to state, and easier to prove.
So you may actually agree with a lot of what I'm saying, but your arguments to support that claim appear, at least on the surface, to be wrong. It may be that you have some understanding of these "paradoxes," but your other statements suggest that your inderstanding is not that of current or classical mathematics.
As a closet constructivist I feel compelled to point out that 2^N certainly has a more complicated internal structure, but "bigger" is entirely a question of interpretation.
There are a countable number of possible constructions of something in 2^N, and therefore in the traditional interpretation of mathematics all but a countable number of members of 2^N are unconstructable. However a constructivist looks at this and asks what sense it makes to talk about the real "existence" of mathematical objects that cannot be represented in any way, shape or form, even in principle? To a constructivist these don't really exist at all, so there really can't be more of them.
Incidentally the classic picture most mathematicians have of the infinite cardinals being nicely sorted by size depends on the axiom of choice. Specifically Friedrich Hartogs proved in 1915 that if all infinite sets are comparable in size, then the axiom of choice holds. The reverse statement is also true, and is easy to prove from the well-ordering principle. So if you want to be a classical mathematician but are heretic enough to doubt the axiom of choice, then again the simple picture people have of infinite cardinals breaks down.
In this way, infinity of any form doesn't actually exist.
Add to the mix the axiom of infinity and it then does exist, but we're close to the constructivist's world.
However, having said that, there's a lot that I do that relies, underneath, on having the more usual model. Allowing the declaration of the existence of all elements of 2^S for any set S is just too useful not to explore. That's the world I usually inhabit.
However, it's not clear that getting into those discussions will, in any way, help people to understand the more "regular" viewpoints and reasoning. Encouraging people's intuition can be good, but sometimes it simply prevents them from reasoning clearly.
And perhaps that's the most easily defensible aspect of the usual idea (within mathematics) of infinity. Dealing with it forces you not to take things for granted, and to reason carefully about properties. Discovering that 1+aleph_0 = aleph_0 = aleph_0+1 from first principles is useful. Then finding that 1+omega != omega+1 is useful.
Mathematical logic and reasoning is so useful, we need places to exercise it. This is one, there are others.
But personally, I like the idea that there are different sizes of infinity, and the resulting implications such as the Banach-Tarski theorem, the existence of uncomputables, and the tension between Zorn's Lemma and the well-ordering principle.
Although mostly it doesn't really matter.
On the other hand, it is also rarely necessary.
1) Clever little things
2) Long, hard slogs
Cantor's proof is a clever little thing. These sorts of proofs can be daunting at first but they become easier, even delightful, as one becomes more familiar with their kind. If you studied the foundations of math in school for as long as an American student takes English literature courses then these proofs would be second nature to you. They aren't really hard, just written in a language unfamiliar to most people.
Then there are the long, hard slogs. Lemma after lemma, with no clue as to where the author is leading, until finally, after an exhausting march, he bludgeons the reader into accepting his conclusions. These are pretty hard for anybody to understand because you can't hold the whole thing in your head. You just have to convince yourself of the truth of each step.
For Cantor, I would imagine that the formal proof was simpler to write out using the binary representation. Just a guess.
In a class I took with my thesis advisor, he explained the Hilbert Hotel similarly, and it was as entertaining then as this is now.
Every time we expect something infinite in physical phenomenon, it turns out to have some sort of Plank-like constant that compartmentalizes the effect at different levels to avoid that result.
Which is good, because a universe where infinity existed would most likely not be stable enough to support life. I understand that mathematics isn't simply about physical phenomenon. I understand the beauty of i, for example. But I think that screwing around with the bizarre aspects of a concept like infinity is a huge waste of time.
Maybe for you, maybe not for someone who enjoys doing it.
Imagine defining a function in terms of itself! That's circular and you can always get away with just using a loop instead.
Well, it turns out recursion often works and is very useful. Just like infinity.
I meant infinity, not the limit of infinity.
I have no problem with the concept of infinity in terms of limits, but that's not what this article is about. It's about the concept of infinity itself. And that is what I disagree with.
It's playing around at the limits of human reason. I don't see it as any different to most sports, music, and art. All are appreciated by different groups of people but, ultimately, "useless" if you do not value the enjoyment gained from exploration for its own sake.
Not knowing that there is an unlimited number of numbers becomes a problem. Trying to do any math without allowing that between any two numbers there's another number is pretty difficult.
There are people who derive a mathematics without the axiom of infinity, but it's weird, distorted, unnatural and doesn't turn out to be as useful as often as the regular variety of math.
Feel free not to believe. Just don't expect to do much advanced math.
Oh, and if you use proof techniques in programming, such as loop invariants or transformation theory, you are using infinity, you just didn't realise it.
I have no problem with the concept of infinity in terms of limits, but that's not what this article is about. It's about the concept of infinity itself. And that is what I disagree with.
Because, in part, you can't have the one without the other. You can't talk about limits without having a model for infinity. They are inextricably entwined, each idea leading to the other.
It's sounds to me like you've simply been put off by loads of badly written Pop-Math articles that attempt to boggle and astound, without showing the real truths as to why the whole subject is really, really cool, and simultaneously, useful and practical.
Brian Clegg's book on infinity gets good reviews, but I haven't read it. I'm thinking of starting a blog series on it, but with everything else I have on I may never really get to it.
But as I say, in your specific case it really depends on your background. If you're really interested perhaps you could email me more details on what you've studied and what you've read, and I can write some brief articles by way of reply.
How about the number three? Does it exist? Or the square root of 2? I can't touch it, that's for sure.
People don't want infinity per se. We want a number system where you can do nice things to numbers and get out other numbers... and have nice theorems like "a continuous graph that goes from negative to positive must cross the zero line somewhere"... and then it turns out that the number system that fulfills our need consists of infinite sequences of digits, and finite sequences won't suffice.
I said, quite clearly, "I understand that mathematics isn't simply about physical phenomenon. I understand the beauty of i, for example."
You mention 3 and the square root of 2. You plug both of those in to: (x == x+1) and you'll get an inequality. Not so with infinity.
Again, Rider has set me straight that I need to learn more. I can believe that, but when I see how often in the physical world we creep up on infinity only to have it not exist, I'm definitely skeptical about plugging in infinity into equations like x == x + 1. It seems it doesn't make any sense at all.
Allow me to illustrate. Take the following problem stated in school geometry terms: A square is cut into triangles of equal area, now prove that their number is even. Interestingly, we don't yet know any elementary solution to this problem that can be stated in school math terms. The only solution known was discovered in the 1970s and relies on something called "p-adic numbers". What are those p-adic numbers you ask? They are weird number-like things that have infinitely many digits to the left of the decimal point. Freaky constructs without any counterpart in the real world, but you can still add them, multiply them and all that. Do such things "really exist"? Don't know, don't care. But they helped us solve a difficult problem and that's all that matters.
Edit: but that still seems to me to reflect infinitely, the adjective, as opposed to infinity, the noun. Still, thanks.
or its even stricter cousin, which not only denies the meaningful existence of infinity, but also the meaningful existence of technically finite but infeasibly large numbers: http://en.wikipedia.org/wiki/Ultrafinitism
That said, I will point out that math is essentially a game played by certain rules known as "axioms", and there is no one true set of axioms. If you choose one set, you get infinity; if you choose another, you don't (but you will pay for that). You can't say which is "more right" or "more wrong" without bringing other subjective standards into it, each of which may have their own utility. While "corresponds to the physical universe" is definitely one useful such standard, it is still not the only one.
Do read up on the limits of computable numbers though; they are interesting, but it's not a free lunch.
Hilbert's hotel is more of a set-theory problem than a calculus or geometry problem, so it's best to think of the set of rooms having infinite cardinality. A set of countably infinite cardinality (the guests) is of the same cardinality when another (finite or) countably infinte set is added, so we can develop a one-to-one correspondence between it and another set of countably infinite cardinality (the rooms). Thinking about it this way demystifies the problem a little.
Edit - I'd love to know why someone downvoted this.
If you got infinity people on infinity busses, you would only have infinity people, not infinity squared. The moment when you close down the first bus and start with person 1 on bus 2, this person should already be on bus 1, or else must bus 1 be finite.
That's how I imagined the busses to be; two non-overlapping infinite sets. No one from the first bus was also on the second.
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gran...
relates one method of numbering passengers on the buses to count them (which I found in another source,
http://faculty.cua.edu/glenn/187f09/hilbert_hotel.pdf
which I used to teach my elementary-age class last Saturday the same trick), which shows that countably infinite buses with countably infinite passengers in each bus can still be accommodated by Hilbert's Hotel, even if all rooms are occupied when the buses arrive.
Not only that but the obviousness of "and so on" is often elided. "And so on" assumes unbounded time and/or resources and also assumes you know intimately the (undoubtedly recursive) algorithm and can codify it. These proofs often blatantly ignore that _in practice_ someone or something must perform this algorithm. (Er, not sure if I used elided correctly back there.) Anyhow, if you have an innate distaste for these types of proof then rest assured that you are not alone.
You may check out the esoteric philosopher René Guénon, R. (1946) The Metaphysical Principles of the Infinitesimal Calculus http://books.google.com/books?id=9KyLPwielTEC
Also check out intuitionism by keerazy Dutch mathematician L.E.J Brouwer which asserts that we must essentially modify some preconceived logical tenets/laws in the face of infinity.
See also intuitionistic logic, intuitionistic type theory and constructivism (mathematics). http://en.wikipedia.org/wiki/Constructivism_(mathematics)
Apologies if you were aware of all this already. I have found it very helpful to respect my suspicions. Regardless of what people may tell you, this stuff is neither simple nor straight-forward once you start thinking deeply enough about the minutiae.
I've never liked people who don't understand math and dress it up with philosophical gook. Math is a practical matter, not high philosophy. If you don't know math, your bridges will fall down and your airplanes won't fly.
The reliance of analysis on Cauchy sequences is not "philosophical gook". zemaj may be naïve in her conclusions (or at least unaware of just how much mathematics one must give up when one takes a constructivist view), but she is hardly alone in her hesitancy about infinity. I wouldn't be sure what to make of the claim that Brouwer, Markov or Martin-Löf didn't know mathematics.
Equally if all you care about is working out if your bridge is going to fall down, you don't really have any need for infinity. In fact you can spend an entire career using math to do all kinds of really awesome things without ever actually understanding math. I'd say that a good 95% of engineers fall into that last category, and I'm certainly not worried about driving over bridges because of it.
No infinity required.
The point, though, was that the behavior characterized by the notation is perfectly reasonable even with finite domains (even if the specific notation is not, which was a fatal flaw).
[1] See "BBC: Dangerous Knowledge" e.g. http://www.abdn.ac.uk/modern/node/164
http://news.ycombinator.com/item?id=797723
http://news.ycombinator.com/item?id=121063
http://news.ycombinator.com/item?id=101255
Frankly, it's a load of crap, and shame on the BBC for having produced it. Contemplating the mathematics of infinity didn't drive them insane, they were already troubled. Hundreds of thousands of mathematicians happily deal with infinity on a daily basis.
What a crock.
"No one shall expel us from the Paradise that Cantor has created."
How can any point in an infinite set be reached in a finite number of steps? For that to be the case, isn't the set in question necessarily a finite set between 0 and infinity?
on a more sensational note, also note that wallace and cantor, both geniuses, also both committed suicide.
http://www.cosmolearning.com/documentaries/dangerous-knowled... is wrong