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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?

Arun Kumar C S
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  • You're going to have to qualify what you understand by "infinity" before anyone can try and answer your question I'm afraid. Do you know that there is more than one "infinity" and how to categorise them? – postmortes Mar 29 '21 at 11:36
  • See Infinite sets for examples. – Mauro ALLEGRANZA Mar 29 '21 at 11:37
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    Relevant: MSE's first question. – Theo Bendit Mar 29 '21 at 11:39
  • I just want to know if there are more points in 2d than in 1d. Now it's a yes or no question. – Arun Kumar C S Mar 29 '21 at 11:39
  • Is the set of points or the set of lines on a plane "larger"? – Mauro ALLEGRANZA.. This is a helpful topic – Arun Kumar C S Mar 29 '21 at 11:42
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    @ArunKumar The word "more" is a little thorny when dealing with infinite quantities. The generally accepted answer in the mathematics community is no, the two spaces have the same number of points (for the usual purposes). – Theo Bendit Mar 29 '21 at 11:45
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    "larger" for infinite sets does not work the same way compared to finite ones. There are infinite sets one of which is a proper subset of the other ($\mathbb N$ ve $\mathbb Z$) and they are "equally large" in the strict mathematical sense. Sorry, if this sound unnatural: it's mathematics... – Mauro ALLEGRANZA Mar 29 '21 at 11:58
  • You should google for the concept of "numerocity" but I know of no published attempt to generalize this to uncountable sets. – Anixx Mar 29 '21 at 12:29
  • @MauroALLEGRANZA $\mathbb{Z}$ and $\mathbb{N}$ are NOT equally large, they have the same cardinality, but different numerocity. Numerocity is a finer measure of size. – Anixx Mar 29 '21 at 12:31
  • @MauroALLEGRANZA the linked answer seems about a different topic completely. – Anixx Mar 29 '21 at 12:41
  • @Anixx. I have edited my answer so that it is more explanatory. – Rounak Sarkar Mar 29 '21 at 14:23
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    @Anixx I'm really curious. I looked up this numerosity thing, and it appears to be an alternative to cardinality of countable sets, invented in 2003, but one that is independent of ZFC. What makes you so adamant that this is the measure of size that the asker is asking about? – Theo Bendit Mar 29 '21 at 15:15
  • @TheoBendit I would also ecommend to read this: https://mathenchant.wordpress.com/2018/09/16/a-new-game-with-infinity/ As to why I guessed that the asked was asking about numerocity, is because the numerocity of a subset is always smaller than the numerocity of the whole. – Anixx Mar 29 '21 at 15:18

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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"...

Dávid Laczkó
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