For some time now, I thought I understood infinity. On *Wayne’s World * a lá SNL, they had a joke. “Infinity plus one”. The joke being, you say infinity. But I will up your statement even by one.
When I first heard this joke, I thought it was ridiculous. Infinity plus one, is still infinity, I would tell myself. But now I am not sure.
I know they say that there are more irrational numbers than rational, even though both are by definition, well, infinite. And this partly Illustrates my confusion.
Anyway, I know that question has already been asked, probably many times. So I will ask another one I have wondered about.
Does a line have twice as many numbers as a ray?
Think about it. A line goes on to infinity in both directions. A ray just one.
I hope the answers to this question will partly help some of my confusion–and hopefully others as well.
It is possible to match up the points of a line one-to-one with the points of a ray, so that there are none left over. In this sense, a line has as many numbers/points as a ray.
It’s also possible to match up the points on a line segment with the points on a ray or line with none left over. But, as you know, you can’t match every real number to an integer. There will be an infinite number of reals left over.
If Garth and Wayne are mathematicians then it makes (or can be made to make) sense.
When you count numbers in a sequence, you start with 1, 2, 3… But how do you describe the sequence of all the natural numbers? That is usually denoted by the letter ω. What comes after ω? ω+1, naturally. Both are infinite ordinal numbers, but they are not equal. Then come ω+2, ω+3, …, ω+ω, ω+ω+1, …, ω[sup]ω[/sup], …, and pretty soon Wayne and Garth will need to resort to increasing mathematical sophistication to continue on to ω[sup]ω[sup]ω[sup]⋰[/sup][/sup][/sup] and far, far beyond.
Now you are thinking of the concept of mathematical cardinal numbers, which answer “how many?” or “who has more?” by matching up elements. It is absolutely true that a line has twice as many numbers as a ray. But twice any infinite cardinal number, possibly assuming the truth of the Axiom of Choice or something, is always the same cardinal and no bigger.
ETA the irrational numbers are uncountable, but the rational numbers are countable, which makes a difference.
Here’s a way to visualize different orders of infinity.
Imagine some set like the set of integers or the multiples of five or the prime numbers laid out on a line. You’d have a line with a bunch of points on it. The line would extend to infinity and you’d have an infinite amount of points on the line.
Now imagine taking any segment of that line. No matter how long the line is it contains a finite amount of numbers; you could count all the members of the set that are contained along this segment of the line.
Now imagine some set like the set of all rational numbers laid out on a line. This line also extends to infinity and has an infinite amount of points on it.
But here’s the difference. Take a segment of this line and you’ll find that there are an infinite amount of members of the set contained within this finite segment of the line.
See the link in what you quoted, or here (especially page 3). Even after every one of the infinitely many integers has been matched with a real number, there must always be some left over.
Everything you said is correct, but it’s a very bad example, because the rational numbers are countable, and therefore not a different order of infinity from integers or multiples of integers.
This whole post confuses me to all heck. The cardinal numbers are the ordinary counting numbers, but the ordinals show place in a count. Among the numerals 1, 2, 3, 4, 5, 6, 7, 8, three is the third number. Cardinal is the ordinal number. I don’t get what you say about “Both are infinite ordinal numbers.” You can’t have infinity plus one as an ordinal.
It also is absolutely not true “that a line has twice as many numbers as a ray.” They have the same cardinality, i.e. are both uncountables infinities and exactly the same size. This is the first and most important thing to understand about infinity. A line segment of any size has exactly as many points as any other line segment. And to every infinite line. And to every three dimensional space. An inch contains the same number of points as the universe. And as every higher dimensional object. But the points of a line are a different infinity than the counting numbers and the rationals.
I’m also pretty sure that the Axiom of Choice plays no part in this.
A line DOES have twice as many numbers as a ray. AND it has exactly as many. The statements are not contradictory.
And the Axiom of Choice (AC) certainly DOES play a role. This paper asserts that
[m][sup]2[/sup] = m, whenever m is infinite
is equivalent to AC, and that 2m = m is a weak form of AC.
You need to know about Cantor’s Proof of the Uncountability of the Real Numbers:
These are each proofs of this theorem. Can someone else recommend other places where this theorem is explained well? You need to understand this before we can begin to answer your question.
The AC implies it: There’s a result of Tarski giving a bijection m = m * m in this case, so m = m + m follows from the Cantor-Bernstein theorem. On the other hand, there exist models in which m = m + m holds for infinite m but AC fails. (The construction is messy and technical, and it’s not very enlightening.)
But you can’t ever find one. If you say “Well, your real number corresponds to 1972363”, then I’ll say “No it doesn’t; my number differs from the one that corresponds to 1972363 in the 1972363rd decimal place”. Likewise for any other number you claim. No matter how many numbers you put on the list, there will always be some real number that is nowhere on the list.
That said, cardinality isn’t the only notion of “size” in mathematics. You can also say that one set is smaller than another if one is a proper subset of the other, for instance, and a ray is a proper subset of a line (of one line, at least). That’s pretty clear, but unfortunately it also leaves a lot of pairs of objects that you can’t compare. You can say that one set has a greater measure than another, and hence that (for instance) a line segment two inches long is bigger than a line segment one inch long (though that doesn’t do any good for sets of measure zero or measure infinity, of both of which there are plenty of examples). I’m pretty sure that for any space of sets, it’s always possible to come up with a well-ordering of them, such that given any two sets, you can always say which one is greater, and where a > b and b > c implies a > c… but there’s no guarantee that that well-ordering will be at all intuitive.
No, and the proof is Cantor’s diagonal argument that was already cited several times. The core of the argument is actually pretty simple. Any real number can be represented as an infinite sequence of digits. Simply put, Cantor showed that any infinite enumerated set of such numbers must be incomplete because you can always create a new number that can’t be found in the set, by constructing a number that differs from the first one in its first digit, from the second one in its second digit, and so on, so it must be different in at least one digit from any of the existing numbers.
You can of course add the new number to the set, but then the diagonal argument works again, and this repeats infinitely. The implication of this is that the set of real numbers is an uncountable infinity.
This is not true for either countable or uncountable infinities.
There are not twice as many integers as there are even integers. They are in a one-to-one correspondence.
Similarly, take the points on the number line from - infinity to plus infinity. A ray is the equivalent of starting at 0 and extending to infinity. The number line is therefore “twice” the ray, but, again, they are in a one-to-one correspondence.
Well, that’s the thing about math. If you think it’s wrong, then you should be able to point out what’s wrong with it. And sure, doing math with infinities is silly: I’ll grant that point. But silly isn’t the same thing as wrong.
“Countable” means “can be put into one-to-one correspondence with the counting numbers,” i.e. the positive integers.
Two infinite sets A and B are said to have the same cardinality if their elements can be put into one-to-one correspondence with each other. But the cardinality of A is said to be greater if B can’t be put into one-to-one correspondence with A, but only with a subset of A.
