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"My understanding of a line is that it is delimited by two points"

That is not how Euclid defined it and how it is still seen in geometry today. What you describe is called a (line) segment (http://en.wikipedia.org/wiki/Line_segment)

"but does not contain any points"

Lines extend indefinitely in two directions (if you go past Euclidean geometry, that 'indefinitely' changes meaning a bit)

One talks of a point being _on_ a line in geometry. 'contains' is something from set theory: "the set of all points on line l contains point P" is a perfectly valid expression (but "P is on l" is way shorter)

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Excuse me. Of course. I was using line and line segment interchangeably there. Which I should have not been doing if I am aiming for clarity but I think my point (ahem) applies to line segments and lines that extend indefinitely in one or two directions. Presumably people will contend that even a line segment "contains" an infinite number of points. But if points have zero extension then even an infinity of them cannot sum to anything greater than zero. So I ask you again, does it make sense to think of lines (or line segments) as composed of points, I reckon it does not.

> But if points have zero extension then even an infinity of them cannot sum to anything greater than zero.

Can you make this rigorous? Because using the standard definitions, this statement is not true. It's true that a countable number of points must have total length zero (and you can even give a rigorous proof of this) but not necessarily true for a non-countable number of points. The study of "lengths of sets of points" is called measure theory.

I think it is unnecessary, however, to bring in the whole concept of length when defining lines. For example, we could simply define a line as a set of points obeying some special properties.

"But if points have zero extension then even an infinity of them cannot sum to anything greater than zero."

Infinities are weird; anybody who wants to learn math has to accept that. 0.99999… does equal 1, there are as many even numbers as integers, etc. these things are 'true' not because they make sense initially, but because they make the most sense of all the other things we have thought of so far. Similarly, a set of Aleph-0 points can completely cover a line.

"a set of Aleph-0 points can completely cover a line"

Aleph_0 is the cardinality of the integers. I don't think that'll cover a line. For that, you need the cardinality of the reals, C, which may or may not be Aleph_1.

OOPS. Thanks

Infinities are weird.

Perhaps your intuition changes if you think about the line (in a plane) coinciding with the x-axis. Don't you think it is reasonable to define that line as the set of points (in the plane) having y=0? Also, should that line not be identical (isomorphic) to the set of real numbers - which is just a set of numbers? And should not all other lines in the plane be identical (isomorphic) to the first line?

Would it make a difference if you substituted "infinitesimal extension" for "zero extension"?

That's the thing though. As I understand it, or as it's said to be: points have zero extension. So, no amount of points, not even an infinity of them could ever have extension. But, it should make sense for a line to be composed of entities with infinitesimal extension as you say. I have seen the term linelet used before for these entities.

Charles Sanders Peirce said in 1903, ”Now if we are to accept the common idea of continuity […] we must either say that a continuous line contains no points or […] that the principle of excluded middle does not hold of these points. The principle of excluded middle applies only to an individual […] but places being mere possibilities without actual existence are not individuals.”

> That's the thing though. As I understand it, or as it's said to be: points have zero extension. So, no amount of points, not even an infinity of them could ever have extension.

This isn't true for an uncountably infinite set of points, assuming by 'extension' you mean what is usually called 'meausre' in modern mathematics. Modern theory is perfectly fine with saying that a line of nonzero length contains an infinite number of points of zero length, and trying to draw on Euclidean notions definitions of 'point' and 'line' to find conclusions about real analysis is going to be unhelpful.

I'm not sure what that Peirce quote is trying to say.

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