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A little dialogue aiming at explaining the question:

A - What is a line ?

B - A set of points that has no width, no depth, but has length, " a line is a breadthless length" ( Euclid, Bk1, Df2)

A - What is a point?

B - Something indivisible, "that which has no part" ( Bk1, Df1) says Euclid, no "extension".

A - How could a line be extended in length if its parts have absolutely no length? For, having no part, points certainly have no length either.

B - I said a line is a set of points. But did I say that these points were the parts of the line?

(The context of this question is basic geometry. )

My question: how to caracterize the mistake made by person A? is person B right when she explains the mistake in terms of membership/ inclusion confusion?

Can one clarify the loose expression "being made of points" by saying (1) yes a line is made of points as elements ( = members) , but(2) it is not made of points as parts ( the parts of the line being not points, but subsets of points)?

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    What you're asking is somewhat imprecise, but in any case, I think what you're looking for is the notion of a topology on the set of points in the line. It's the topology that makes the points 'touch'. – Clive Newstead Apr 11 '19 at 17:19
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    "a line has to be made out of something" No, it doesn't. In modern axiomatic treatments of geometry (specifically with Hilbert's axioms), points and lines aren't defined in terms of anything else, they just are. A point can lie on a line, but a line is in no way a collection of points. – Arthur Apr 11 '19 at 17:35
  • @ Arthur. - I have no knowledge of advanced geometry, my question is asked at the most basic level. I thought that some set theoretic notions were used in basic geometry, for example: " The circle C with center O and radius 5 is the set of points in plane P lying at a distance of 5 units from point O". Is this correct? –  Apr 11 '19 at 17:40
  • I'm not saying you have to do it my way. I'm saying that a line doesn't have to be made of anything. And circles are different. Yes, that is how I would define a circle. And if your interpretation is more along the lines of what Clive says above (and lines indeed are collections of points), then while no two points touch, the whole collection of points is indeed connected. There is no issue here, only lacking intuition about what can happen when you put infinitely many points in close proximity to one another. – Arthur Apr 11 '19 at 17:45
  • @ Arthur - Just to explain the motivations of the question. I do not claim there is really an issue. To the contrary I'd like to know whether set theory ( as I suspect) gives us an illuminating tool to show there is not issue ( although even the greatest mathematicians in the past as Pascal or Leibniz thought there was a kind of " mystery" in the " continuum" problem). –  Apr 11 '19 at 17:58
  • @ Arthur - Would you say the question of " connectedness" relates to the distinction between membership and inclusion? Could one say that connectedness is a property of sets ( but not of the elements of these sets)? And that the alledged contradiction I refer to in my post comes from a confusion between properties of points and properties of sets of points? –  Apr 11 '19 at 18:01
  • Can you define what you mean by the "membership vs inclusion" problem? – Sort of Damocles Apr 11 '19 at 19:25
  • @ dbx - I don't consider the set theoretic distinction between membership and inclusion as a problem in itself; rather, I'w wondering whether it could not provide an illuminating solution to the problem I stated regarding the composition of a line : namely the somewhat naive ( but legitimate ) question , " what is a geometric line made of"? –  Apr 11 '19 at 19:30
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    This is more a question of philosophy than math. Modern mathematics would reject both (1) and (2) as meaninglessly vague statements unless you provide precise definitions for all the terms involved ("point", "line", "produce", "touch", etc). – Eric Wofsey Apr 11 '19 at 20:27
  • @ Eric Wofsey - I've added some definitions from Euclid and a little dialogue aiming at showing how the question could arise in a natural way ( at the elementary level). –  Apr 11 '19 at 21:29
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    No, this doesn't solve the problem, since it doesn't explain how it is that a square can be made up of lines. A line segment is a subset of a square, not an element of it. – Jack M Apr 11 '19 at 21:31
  • @ Jack M.- You're right, it would be false to believe that the distinction elements/part is repeated at every level ( line, surface, solid, etc). At every level, the only elements are points. All the other objects in the hierarchy are sets of points, and the only relation that holds between these objects is the subset/ superset relation. A squareis a set of points in the same way as a line is a set of points. –  Apr 11 '19 at 21:49
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    Relevant earlier question: Should we think of lines as sets of points? – hmakholm left over Monica Apr 11 '19 at 22:22
  • @ Henning Makholm - Thanks for this interesting link. –  Apr 11 '19 at 22:25
  • Also relevant : https://math.stackexchange.com/questions/1083841/if-point-is-zero-dimensional-how-can-it-form-a-finite-one-dimensional-line –  Apr 12 '19 at 00:43
  • If you use @ to notify people, it's best if don't have a space between the @ and the name (also remove all spaces in the name). If you have spaces we don't get notifications. – Arthur Apr 12 '19 at 07:03
  • @Arthur-Did not know that, thanks. –  Apr 12 '19 at 13:24

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