Thread: Different kinds of infinity?

I recently watched a documentary that introduced the Continuum Hypothesis (nothing in depth at all) and George Cantor, the famous mathematician. They discussed the notion of different kinds of infinities, but didn't get into the details or explanations. I assume this is because this aired on public television, and the technicalities would go over the heads of... everyone. I don't expect I'd understand right off the bat either.

In any case, the video described a circle in a new way to me. A circle was superimposed on the screen, and the narrator would draw a triangle, a square, and so on inscribed within the circle, and then went on to explain that a circle was essentially a shape with an infinite number of angles if all of the side and angles were the same. After thinking about it for a moment however it also occurred to me that circles have perfect curves, and no sides as can be distinguished be being separate from one another by any angle. A circle has infinite angles, but it also has zero angles? Is something like that theoretically possible? Cantor went insane and was put into an asylum several times over the course of his attempts to prove the Continuum hypothesis. The documentary touched upon contradictory elements in his work.. Again though, there isn't much in terms of specifics.

To conclude: I want to know more about different kinds of infinity, and whether or not infinity can be equal to zero in some cases.

PS- I found the documentary on page 65 of the Abstruse Goose webcomic, aptly titled, "The Cantor Madness."

PPS- I have some understanding of countable and uncountable number sets.