Infinite number of points in the Sierpinski Triangle

I have basic background in mathematics (Linear Algebra, Calculus) and I've been reading up on fractals, because I find them fascinating. I can't understand one thing in basically all of the fractals created using the iterated function systems.

For example, in the Sierpinski Triangle it can be easily shown using a limit, that after an infinity of iterations there will be nothing left of the triangle, and yet I will still have an infinite number of points. The same thing happens with Cantor's Comb, etc. How can I remove all of the area and still have an infinite number of points?

Could you help me understand this? Please keep in mind that I'm not very proficient in mathematical logic.

Thanks,

Solution 1:

It can sound a little confusing at first, but remember that there are also infinitely many points in a line, that doesn't have any area either.