Sizes of infinity

Question

I was just thinking about infinity (as you do) and thought the following.

"There are infinitely many reals in the interval $x\in[0,1]$ and an 'equal number of reals' $x\in[1,2]$, so there are 'double the number of reals' in the interval $x\in[0,2]$ in comparison to $x\in[0,1]$"

Now I understand that the terminology I have italicized may be incorrect for whatever reason. I have a few questions regarding the statement.

1. Is there a branch of mathematics involving the 'measuring' or 'comparison' of infinities? My gut feeling is telling me something to do with set theory.

2. What key terms should I look up on Wikipedia regarding this?

3. The parts in italics, what is the 'correct' way of saying this, assuming it is incorrect to say it with that precise wording?

4. Using mathematical notation, how do we express the idea of one infinity being 'twice' the size of the other infinity?

Answer 1

1. Yes. Set theory.

2. "Countable set" would be a good starting point on Wikipedia.

3. "Equal number of" is fine. The more formal term is that two sets have the "same cardinality"

   "Double the number" is wrong.

   You have to think what "equal number of" means for infinite sets. The answer is that there is a one-one correspondence between them (usually called a "bijection"). For example, $n\to2n$ is a bijection between the natural numbers and the (positive) even numbers.

4. Well "double" is not a useful concept for infinite numbers. Because if you have an infinite set and compare it with the result of "doubling it" you find they are the same size. More generally, what happens if you "multiply" or "add" infinite numbers is known as "cardinal arithmetic".

Answer 2

A way to write this using cardinal numbers is $$|[0,1]| = |[1,2]| = 2^{\aleph_0}$$ And $$|[0,2]| = |[0,1]| + |[1,2]| = 2^{\aleph_0} + 2^{\aleph_0} = 2^{\aleph_0}$$ Note the last equality seems irritating at first, but know that $|\mathbb R| = 2^{\aleph_0}$ where $|\mathbb N| = \aleph_0$. Two sets have the same cardinality if there exists a bijection between them.

From a measure-theoretic point of view, if we use the standard lebesgue measure, things are different, but may seem more natural to you: $$\lambda([0,1]) = \lambda([1,2]) = 1$$ And $$\lambda([0,2]) = \lambda([0,1]) + \lambda([1,2]) = 1+1 = 2$$ Note however that $\lambda(\mathbb R) = \infty, \lambda(\mathbb N) = 0$.