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Since a point in mathematics is dimension less, how can we get an object of 1 dimension by combining dimension less objects ?

Further more, into how many parts can we divide a line If we can divide a line segment and obtain an infinite number of points, doesn't this imply that the line segment should have an infinite length. This sounds really similar to Zeno's paradox but l want to understand how mathematicians treat the problem.

Finally if we can divide the line segment into an infinite numbers of points, is the set of points countable or uncountable ?

How does Cantor set fit into all this ?

Aristotle Stagiritis
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  • You may find answers to all your questions if you read about cardinality and especially cardinality of the continuum – Anatoliy R Nov 27 '19 at 03:56
  • And https://math.stackexchange.com/questions/2417029/how-can-points-that-have-length-zero-result-in-a-line-segment-with-finite-length?noredirect=1&lq=1 and https://math.stackexchange.com/questions/1936865/what-is-the-length-of-a-point-on-the-real-number-line?noredirect=1&lq=1 – rschwieb Nov 27 '19 at 04:00
  • I have edited the question – Aristotle Stagiritis Nov 27 '19 at 04:25

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When we say that a point is dimensionless, we mean that if your world consists of only one point then you do not need any information to describe your world.

If your world is one dimensional then you need a reference point and for every other point in your world you just need one information to locate where you are in comparison to the reference point.

For a two dimensional world you need a set of two information to locate every point and so forth.

The points on a line are not countable as it is proved in real analysis and yes you can divide a line segment into infinite parts without going to infinite length.

For example you know that $$\frac {1}{2} +\frac {1}{4}+\frac {1}{8}+....=1$$ so you can add infinitely many positive real numbers and get to $1$

Mohammad Riazi-Kermani
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