Ok, lately I have been thinking a lot about one idea that has been bothering me since first I learned about lines and points.
I understood that:
A line has no thickness, is straight and it is extending infinitely in "both" directions.
And that a point is an exact position without size.
Synthesizing this I thought of a line as an infinite serie of points. (linear set if you will)
But now I came up with the idea that actually, there is no "both directions" if you don't choose a point from the line as a reference from which those directions emerge.
Thus I reasoned that actually, taking any single point from the line would allow me to end up with the same. Thus allowing me to create a y axis from which to make it extend to infinity in both sides.
And then it hit me, by just choosing a point you can distinguish between "1" or "2" infinities.
Both infinite, yet choosing a point "divides" (I know I can't do that) the original one in seemingly 2 different infinites. Which actually "both" have the same properties as the one we started with.
So if I choose 0 points on that line, that line is composed of an infinite amount of points.
- 1 point to get 2 infinite lines.
- 2 points and we have 3 infinite lines
- 3 points and we have 4..
- 4 points and we have 5..
So now let's say I take an infinite amount of points,
I would still have an infinite amount of lines of infinite length between an infinite amount of points.
All with the same properties as the first infinity we started with.
But now a question bothers me a lot:
Could I tell the difference between a gap of 0 points or a gap of an infinite amount of points ?
I hope someone can point me the way out about this theoretical problem.
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