TITLE: Does a higher dimensional space have more points than a low dimensional space? And in some way are some infinities bigger than others?
QUESTION [1 upvotes]: How many points can exist in one dimension? ANS: Infinite!
How many lines can exist in a two dimension? ANS: Infinite!
Now my question is how many points can exist two dimensions or even higher dimensions?
"Infinite points * Infinite lines" infinity? Or "just" infinity?
Can we say two dimensions contain more points than one dimension?

REPLY [0 votes]: Consider 2 parallel lines in the same plane. How many points can you have on one line? Any point on the other line is not in the set of points on the first line, but in the set of points on the plane. Can those points on the second line be qualified as "more"? I guess yes, and the statement that "two dimensions contain more points than one dimension" should be true, however it is still "just infinity": how many line segments can you have on a line? How many points can you have on a line segment? A line on its own can have uncountably infinite points, and that is still "just infinity", but there is a difference between countably and uncountably infinite. Every line is a real number line, and every point on it is a real number, so the "How many points can exist in one dimension? ANS: Infinite!" statement can be broken down more on its own, but it is "just infinity" at the same time. So even if we "stay" in 1-D we can arrive at this problem, we don't "have to go higher"...
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