Here's how I heard someone formulate the difference between countable and uncountable infinities:

Say you're in Hell, but the devil's a sporting fellow, so he's willing to make a deal with you. He's got a number in mind, and if you can guess that number, he'll let you go free. (You get one guess a day and he promises to never change the number he's chosen.)

Now, if he tells you that he's chosen a positive integer, you know you're in good shape. You just need to start at 0, and guess the next highest integer every day until you're released: you're guaranteed to make it up to his number eventually.

Now, what if he says it can be any integer, positive or negative? Actually, you're still in good shape: start with 0, then guess 1, -1, 2, -2, and so on. You're still guaranteed to hit the right number eventually.

But what if he tells you that it could be any real number between 0 and 1? Can you come up with any method that's guaranteed to go methodically through all the options so that you're sure you'll hit the right one eventually? No, you can't, so this set of numbers is larger. (SG-fan's diagonalization technique proves this.)

In the first two cases, you were guessing from a countably infinite set, whereas in the latter, you're guessing form an uncountably infinite one.

I see what you've done there - very cunning.

Who needs a maths degree?

This reminds me of the proof about hell being endothermic or exothermic.

Yeah, but it's still different, because we aren't talking about finite quantities. With finite numbers, there is a measurable difference between two different numbers. With infinity, there is not, because infinity is not a number, no matter how you look at it. So yeah, real numbers are larger in one sense, but not in a finite sense.

I don't follow your argument, Obi. There are more numbers in the set of reals between 0 and 1 than there are in integers. This is proven.

Yes, but neither of the numbers are finite. You can't take the difference between the two sets because neither set has a real magnitude. One set is "larger" but not in the same way that a finite set is larger than another finite set. The magnitude of the set of counting numbers minus the magnitude of real numbers between 0 and 1 doesn't even make sense because in some sense, infinity doesn't really have a size.

Well, you could still algebraically show that the set of reals on [0,1] is at least X times larger than the set of integers, depending on how deep you want to dive. I could easily represent the size of the set of integers as X and show that the set of reals on [0,1] is at least 2X. So, in that sense you can take the different between the two.

Isn't the set of reals infinitely times larger? Either way you do it, the difference is still infinity. The magnitude of infinity does not really have a meaning in this case. I mean, the set of reals is larger in the sense that there are more of them, but not in the sense that it has a larger magnitude, because infinity does not really have a size.

I'm saying I could show that it is at least twice as large, for the sake of just showing that the set of reals is larger. In this case, I think "taking the difference" is appropriately applicable.

So, forgive me for being dense, for I'm a humanities guy and not a math and sciences guy, but why is counting like such:

• 0.1
• 0.2
• 0.3
• 0.4
• 0.5
• 0.6
• 0.7
• 0.8
• 0.9
• 0.10
• 0.11
• 0.12...

Take a "larger" infinity than this:

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11
• 12...

I would think the former would be "smaller" since numbers like 0.1 and 0.10 are the same value. The only possibility I see is that the former have irrational numbers as well, but I don't know what the "proportion" of irrational to rational numbers would be (if such a thing exists). But whatever, I'm sure I'm being stupid on the matter.

In any case, I don't see how the Devil example works -- can't I "methodically" count using my first method?

I don't think twice wouldn't cut it because you'd still have cardinality. I think I could graphically show that the set of reals is infinity times larger.

Infinity minus infinity has no rational meaning even if one set contains more number/isn't cardinal with the other.

You can count the rational numbers between 0 and 1, but not the irrationals. What comes after 0.000...001? (That number is equal to zero.)

Or is it?

Ah yes, I forgot all the infinite number of .01 and .001 and so on. I guess coming up with a "countable" method to include those would be a bit difficult, which isn't even including irrationals.

I find it easier to accept when I feel maths can be shown visually and/or told that it's vital to making my computer work. Let me know when you all decide to strike up an art, writing and/or media/communication studies debate.

I assume what you meant by 0.000...001 is that there is an infinite number of zeroes with a 1 at the end?

Last I checked, pretend fantasy numbers don't equal zero, or, well, anything. You can't have a 1 at the end of an INFINITE number of zeroes. If there's a 1 at the end, then the zeroes aren't infinite.

I don't see the difference. You're totally boned either way. There are integers so big that we couldn't possibly hope to name them. I can put an arbitrary number of 0's after a one just like I can put an arbitrary number of 0's before a one with a real number. Your example doesn't show and difference at all. I think the so-called 'distinction' is not really one at all.

It's the difference between "eventually" and "potentially never". Huge difference.

Anyways, it's not like it's open for debate; it's mathematics.

And yet you have the same odds of guessing an arbitrary integer as you do an arbitrary real between 0 and 1: 0.

Where did odds ever come into this? It's like Obi's reasoning about one set going to infinity "faster".

The point is that with the integers, if you guess them sequentially, then there is a 100% chance that you'll get it in a finite amount of time. You can't make any such guarantee about the reals, which makes them very different.

Umm wtf? Of course you can!

Not true, and here's why:

Imagine that instead of being from 0 to 1, it's all the real numbers from 0 to 2. You know that these are the same size (both infinite), so it's no problem.

Now say you have a wonderful computer algorithm or something that can generate a list of all the real numbers between 0 and 1, so to get the list that you need you'll just run it twice, once for 0 to 1, and then again but adding 1 to each number.

Say the number you need to guess is 1.5. Will you get to that in a finite amount of time? Because by your claim, you will.

Edit: I realize this reasoning looks really muddled, I'm sure someone understands what I'm getting at though and can make it more clear. This is why I'm not a math teacher

Yes, you'll get in a finite amount of time.

You already explained that with integers you'll get the answer by running through them sequentially, like 1, -1, 2, -2, 3, -3, etc.

You can do the same thing with reals. You just have to assign it a sequence. One sequence I find easy to understand is listing all the possible 1-digit mantissas, then listing all the 2-digit mantissas, etc. So it looks something like this:

0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 (that's all the 1-digit mantissas)
0.00, 0.01, 0.02, 0.03, 0.04, 0.05 ... etc (that's all the 2-digit mantissas)

(note that there's always a finite number of combinations for any length mantissa..)

You'll get the answer in a finite amount of time, and the funny thing is that for EVERY number in the real sequence, I can list a new number in the integer sequence, therefore both sets are the same size and the average time that you'll find the right number is exactly the same.

You forgot irrationals. And repeating rationals.

The cardinality of the set of rationals is the same as the set of integers. This is well known and proven. But the set of reals also includes irrationals, for which there are an infinite number. Therefore the set of reals is infinitely larger than the set of integers.

Say the number you need is pi-3 (0.141592653589793.....)

Assuming a constant speed of generation, how long will it take before your method comes up with it?

The devil wouldn't be able to choose pi because the devil himself doesn't even know its value - nor do we.

Oh I see.

I thought we were having a serious discussion, but if you're just going to troll it then I'm not going to bother.

no I'm serious.

How is Satan supposed to choose a real number for you to guess if it has an infinite number of digits in its mantissa. He would never be able to validate your answer/

And you're acting like I couldn't just do similar bull**** with integers.

I could say that the integer you need to guess is pi's mantissa.

HUURRRR de DURRRRR you'll never guess it, and somehow I'm allowed to ask you to guess it even though I don't know what it is

Well then he can have e then!

But seriously, Vornskr's analogy puts it beyond any doubt for me.

So I guess what I'm getting at is that it's meaningless to ask someone to list even a SINGLE irrational number because all the hard drive space in the world wouldn't be able to hold such a number. You may assume that whatever I was talking about in the thread earlier only applies to rational numbers because at the time I didn't realize what bull**** irrational numbers are. Irrational numbers are quite aptly named and I balk at calling them 'real' because real they are clearly not.

Oy, the idea is that if you start at the integer 0 and go up by 1 each time, you will never miss a number. But using diagonalization, I showed that no matter what numbers you use while counting your reals, there will always be at least 1 number you didn't count.

Oh, and if you don't agree with irrational numbers, then am I to assume you don't agree with infinity either?

No, I 'agree' with infinity. It's a very useful concept. What I disagree with is that it's possible to know the value of an irrational number. And if you don't know a number, then you cannot ask someone to guess it, because you cannot validate his answer. So my problem is with the analogy and not the mathematics.

Oh, ok then

I agree with you on the fact that we can't know an infinitely long number (irrational) And that it makes questions like this really annoying, for the very reasons you said. Though you have to wonder if the devil would know pi infinitely long... hmm

[edit] If you're not bothered by what I said about diagonalization working without regard to stupid irrational oddities, then you don't need to keep reading [/edit] I just want to point you back to the diagonalization proof and mention that it doesn't involve irrationals per se. It's just saying that if you have the same number of infinite numbers between the real set and the integer set, I can still make another real number that's not already counted but you can't make any more integers (since you already used them from 0 to infinity). I prefer that proof to any funky irrational stuff.

Hold on..

now it seems to me that I can always list a new integer between the highest one I already listed and infinity for each new real you may find.

If Satan is thinking of an irrational number, you could suppose that he is computing the digits as you're guessing. Even if you compute the same number at the same speed as Satan, he'll be one digit ahead and always be able to say 'no! not the numer I was thinking of!'

Stretching the analogy a little, but at least it gives you and Satan something to do with your time.

It really is quite simple to understand.

You should refrain from doing so in serious discussions, because the rest of the math world has called them "real" for the past half eon and they fit the definition completely.

Also you can know the value of an irrational number. √29 is irrational but I do know the value. The value is √29.

Hint for Freelancer: just because you can't express a number in decimal notation does not mean it isn't real.

Also I've been waiting all night to break out the proof that sqrt(2) is irrational, it's such an elegant proof; so could someone ask whether all numbers could be expressed as a ratio or something? That'd be great.

Pi wouldn't work for the devil anyway because he's bound by the bible and every good Christian knows that Pi=3!