The set of all circles with different curvatures includes those different circles, whether they intersect some infinitely long straight line, or not.
So the first circle that intersects your line is not “omitted” as you claimed before.
1) This first circle is omitted from the rest of the circles that get the pairs, so there is a pair of points that does not cover the line, simply becuse there are no two circles with the same curvature in the set of all circles with different curvatures.
Glad we could clear that up.
If intersecting, then any intersection is associated to some circle, which its curvature is different than the rest of the circles that are included as unique members (by their different curvatures) of the set of all circles with different curvatures.
Don’t forget there is a whole subset of “circles with different curvatures” that do not intersect your line when those concentric circles are not centered on your line.
Since each circle of the set of all circles with different curvatures is included once and only once as a member of this set, then any unique circle, which does not intersect the considered infinitely long straight line or it is used as a tangent circle with the considered line, its unique curvature is already used, and as a result it does not have a pair of intersecting points, which are associated with it, and this pair does not cover the infinitely long straight line.
How many “circles with different curvatures” do you have? Have you run out for some reason?
You seem to be thinking as if you had only a finite number of circles to use. So tell us Doron how many circles “with different curvatures” are in your “set of all circles with different curvatures”?
Again as noted above: there is a whole subset of “circles with different curvatures” that do not intersect your line when those concentric circles are not centered on your line. That some of those circles do not intersect your line in no way restricts the rest of those concentric circles from intersecting every point on your line. So you still need to identify what point(s) on your line can not be intersected by some member of that set of concentric circles that is not centered on your line.
You’ve gone a long way around once again only to end up right where you were before, unable to identify any point on your line that is not covered by a point and now even just not intersected by a concentric circle not centered on your line.
As a result the set of all points along an infinitely long striated line, is incomplete, even if one of the unique circles is used as a tangent circle (in this case the other point of the potential pair of the tangent circle, is not on the infinitely long striated line).
Doron the fact that one can define a set that includes some, none or even all but one point of your line in no way makes your line incomplete. In fact we can define any number of sets that include all points along your line, the simplest one being, well, your line.
Again this is just your typical self contradictory nonsense: Please identify what member(s) of “the set of all points along an infinitely long striated line” “is not on the infinitely long striated line”.
