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.
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.
What key terms should I look up on Wikipedia regarding this?
The parts in italics, what is the 'correct' way of saying this, assuming it is incorrect to say it with that precise wording?
Using mathematical notation, how do we express the idea of one infinity being 'twice' the size of the other infinity?