That’s like saying two apples have the same number of apples as one apple.
By definition, to add some quantity to some other quantity is to increase that other quantity. So if you’re adding to infinity, you’re increasing it.
An infinite number of centimeters is made out of an infinite number of hundreds of centimeters i.e. meters. This is true. What is not true is that it is made out of the same infinite number of meters. Rather, it is made out of a smaller infinity of meters, specifically, hundred times smaller infinity of meters.
$$ \infty\hspace{0.1cm}cm = \frac{\infty}{100} \hspace{0.1cm}m $$
If you have an infinite number of centimeters and you add to it 99 infinities of centimeters, you get a bigger number, and that number is an infinity of meters.
$$ 100\times\infty\hspace{0.1cm}cm = \infty\hspace{0.1cm}m $$
So, an infinite number of centimeters is clearly smaller than an infinite number of meters.
$$
\frac{\infty}{100} \hspace{0.1cm}m < \infty\hspace{0.1cm}m
\
\infty\hspace{0.1cm}cm < 100\times\infty\hspace{0.1cm}cm
$$
Now, let us return to this proof. The problematic part is:
$$ 10\times0.999\dotso = 9 + 0.999\dotso $$
By writing these infinite decimals as infinite sums, we get:
$$
10\times\sum_{n=1}^{\infty} \frac{9}{10^n} = 9 + \sum_{n=1}^{\infty - 1} \frac{9}{10^n}
\
0.999\dotso = \sum_{n=1}^{\infty} \frac{9}{10^n} \neq \sum_{n=1}^{\infty - 1} \frac{9}{10^n}
$$
Magnus!
Infinity is not a quantity. It is a description of quantity.
You’re still treating it like a quantity.
I think you’re right about that. The two infinitely long sticks are equal only because they’re infinite, which means neither one is longer or shorter than the other, so that would seem to leave “equal”. But I like my other answer to this question better: their equality is undeterminable. Equality of some quantity implies finitude of that quantity (by definition). Infinity is not a quantity. You cannot say two infinite things are equal in the quantitative sense (rather, they are equal in terms of quality–or the property of being infinite).
I don’t get it. I agree that dividing (\infty) by any number is still (\infty), but then where do you get (a < \infty \hspace{0.1cm} \text{ninety inches segments}) and (b < \infty \hspace{0.1cm} \text{hundred inches segments})?
In any case, all this is to show that if you have an infinite amount of something and you have an infinite amount of a bigger something, the infinity of bigger somethings is not greater than the infinity of smaller somethings.
And that means exactly what? What do you mean I am treating infinity like quantity? What does that mean?
On the other hand, considering that addition (and other arithmetic operations) operate on quantities, when you say something like “Infinity + 1 = Infinity” you are treating infinity as a quantity. So I don’t know what to think.
All in all, I don’t understand your objection. So either you have to expand upon it or we have to agree to disagree. Thank you very much.
Anybody ever heard of the Banach-Tarski paradox?
[youtube]http://www.youtube.com/watch?v=s86-Z-CbaHA[/youtube]
He gets into the paradox at about 1:20 but I recommend watching the whole thing anyway since you’re skipping only about 1/20th of to get to 1:20.
The Banach-Tarski paradox says that you can take a sphere and by doing some fancy rearranging and manipulations of the points making up the surface of that sphere, generate 5 replicas of the sphere without adding any additional points. Admittedly, I get lost part way through the video so I can’t claim to understand the paradox, but Ecmandu’s scenario of adding a parallel infinite line next to another infinite line reminded me of it. I thought of this counter-scenario which I think is a sort of watered down version of the Banach-Tarski paradox:
Instead of adding an additional line next to the first one, imagine this: you take every odd point in the first line and move them out of the line. You make a new parallel line out of it. Both lines are still infinite. Now shift each point in each line to fill the gap left from removing every odd point. I hope you’ll agree that this doesn’t make either line shorter. Both lines are still infinite. But then how is this scenario any different than the one we had originally? In this case, we don’t add any points, so we can’t say there are twice as many points. But if this scenario is equivalent to the scenario of adding a second line, how can we say there are twice as many points in the latter case but not the former?
Any two infinitely long sticks are necessarily equal only in the sense that they are infinitely long. That does not mean they are necessarily equal in the mathematical sense of the word (which they aren’t.)
Any two philosophers are necessarily equal only in the sense that they are both philosophers. That does not mean they are necessarily the same person. That does not have to be the case.
You mean to say that, by definition, equality can only exist between two finite quantities? I disagree with that.
To say that two things are equal is to say that they are identical in all relevant aspects. The two things need not be finite at all. Two forests each one of which is made out of an infinite number of trees are necessarily equal in the sense that they have the same number of trees.
Not sure what that means. Certainly, infinity behaves like a number in many ways. For example, you can compare it to other numbers and determine whether it’s greater than or lesser than them.
Dividing an infinite number by any number is still an infinite number in the same way that dividing a finite number by any finite number is still a finite number. That does not, however, mean that dividing an infinite number gives you an infinite number equal in size in the same way that dividing a finite number does not mean you will get a finite number that is equal in size.
(8) divided by (4) equals (2). It does not equal (8). But both (8) and (2) are finite numbers. Similarly, (\infty) divided by (100) gives you (\frac{\infty}{100}). It does not equal (\infty). But both (\infty) and (\frac{\infty}{100}) are infinite numbers.
No, I mean to say quantities are, by definition, finite. Infinity is not a quantity.
While I agree that the two forests are equal in terms of both having the property of being infinite, I wouldn’t say this boils down to quantity of trees. It sounds odd to say, but the forests don’t have a quantity of trees. Infinity is not a quantity.
Yes, you can take the symbol (\infty) and plug it into mathematical formulas without incurring any issues. I can’t think of a scenario in which you can show (\infty) to be less than some number.
I don’t think we can define the “size” of an infinite set. Certainly, we can say an infinite set is greater in size than any finite set, but I think this is a limitation of language. It should not be read to imply that the infinite set has a size. I don’t think infinity behaves like numbers, not in all ways, I don’t think you can divide infinity by a number and get a smaller infinity. That would be like saying if you walk two feet towards the edge of the universe (assuming it’s an infinite distance away), you’re two feet closer to the edge of the universe. If you imagine only yourself in the universe, nothing else, then how is the scenario in which you move two feet closer to the edge of the universe different than if you stay in one place? Similarly, how is the scenario in which you divide infinity by some number different than if you don’t divide infinity by that number? How are the two infinities different?
But then you’re saying (\frac{\infty}{100}) != (\infty). And you’re also saying (\frac{\infty}{100}) is not finite. So it’s not (\infty) and it’s not finite. What’s left?
Magnus!
Let me try to explain this to you a different way then.
Infinity is not a number, it is an operator.
What is plus divided by plus?
It’s a nonsense question. Plus and division are operators not numbers.
You and cantor made the same mistake, you are treating infinity as a quantity. You can’t subtract, add, multiply or divide infinity.
If it’s not finite then it’s infinite. So (\frac{\infty}{100}) is infinite and yet not equal to (\infty). Looks like a contradiction if we assume that the symbol (\infty) and the word “infinite” have the same meaning. But in the context of my post, that’s not the case. The symbol and the word mean two different albeit related things. When I use the word “infinite”, I mean “any infinite quantity in general”. When I use the symbol (\infty), I’m talking about some specific infinite quantity. So (\infty) is infinite and so is (\frac{\infty}{100}) but they are not equal i.e. they are not the same infinite number. This is why James makes use of (infA). When he says (infA), he’s making it perfectly clear that he’s talking about some specific infinite quantity rather than infinity in general.
Let’s say that quantities are by definition finite and that infinity is thus not a quantity. What does this imply? What kind of impact does this have on the current discussion?
Depends on what you mean by (\infty). If what you mean is infinity in the broadest sense of the word, then yes, there are infinities smaller than other numbers. For example, negative infinity is smaller than all positive finite numbers. Then you have infinitesimals such as (\frac{1}{\infty}) which are greater than zero but smaller than any other positive number. Then there are infinite decimals such as the debated (0.999\cdots). These are all, in the very general sense of the word, infinite quantities.
You don’t agree with the claim that a planet populated by an infinite number of organisms has fewer organisms than two planets populated by infinite number of organisms? I think that’s pretty obvious. You don’t have to treat infinity as a number. You can treat it as a unit, as something that you can count. One infinite train + one infinite train = two infinite trains. Two infinite trains > one infinite train. The difference between the two of them is not zero, else they would be one and the same train.
So if you take something and cut it into pieces, you don’t get pieces that are smaller than the thing that you cut? They are the same size? Or maybe you think that sort of operation doesn’t apply to infinity? But because of what? Because infinity is not finite number? But we can say that we’re not dividing a number, or rather, that we’re dividing a number but that that number is one: one infinity. You cut one infinity into four pieces. What do you get? You get four quarters of one infinity. That’s dividing number one. What about dividing things that are not numbers? Are meters numbers? Of course not. And yet, they can be cut into centimeters.
It’s different by the fact that you moved away from your prior position.
[b]
Infinity is not a number but a placeholder for a non finite quantity
There is more than one infinity and no two are the same size
The smallest infinity is that of the standard number line
Infinity plus I = infinity
Infinity minus I = infinity
Infinity divided by infinity = I
Infinity times infinity = infinity
Infinity is in theory 99999 … but it cannot be expressed as such
For that is a number whereas infinity is not a number but a non finite quantity
[/b]
You can quantify operators.
One plus sign plus one plus sign equals two plus signs.
But I am not really sure that infinity is an operator.
No! Plus, plus plus is not a number.
Infinity is not a number.
Centimeters aren’t numbers, and yet, you can quantify them.
Centimeters are units of measurement … operators are not units of measurement.
Anything can be a unit of measurement including things that are infinite.
Operators are not units of measurement.
Hard to disagree with that.
The reason why infinity is not a number is because it does not occupy a specific place on the number line
All the numbers on the number line are both finite and fixed while infinity by contrast is neither of these
Infinity exists as a placeholder because the largest finite number is one that can never actually be known