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Try to come up with a set non-colinear points where NO line passes through two and ONLY TWO points and you'll see the value of the statement.

You may think "I'm sure I can arrange these points in a way where EVERY line will cross three or more points" but you will fail if you try unless ALL points are colinear.

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This is true for finite sets. For infinite sets, the Sierpinski triangle is a counterexample.

> the Sierpinski triangle is a counterexample

How so? It's bounded by the large initial triangle. The line containing any two of the vertices doesn't intersect any other point.

That’s an uncountable set. If we want a counter example for uncountable sets a simpler example is a circular area.

Anyone happen to know if it is true for countably infinite sets?

Just a regular grid?

That's countable you just go in a spiral.

You can always find a line that passes between two points. Why would you even try to find a line that doesn’t pass between two points? What is the difficult part here.

Find one that bridges only two points, no more. Unless ofc you're given a configuration that's obviously impossible: all of them in a line.

The difficult part is that you may think that there's an arrangement where you could force every line to go through 3 or more points. The line extends to infinity, keep in mind.

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