I can't speak to the first and third examples, but the "painter's paradox" is silly, I think, not a paradox at all. It's easier to visualize in two dimensions: consider the function y = 1 / x ^ 2 (y equals one over x squared). We can start from 1 and head out to infinity, and the length of the line will be infinite, but the area under it will be finite. Extrude a fixed amount in and out of the page and we have an ugly three dimensional "horn" with the desired properties: finite volume and infinite surface area. But there's no paradox: you just have to keep painting in thinner and thinner coats the farther out you go. Of course, it soon becomes a tiny fraction of an atom's breadth in thickness, so no real paint would do, but there is no paradox.
You are right, but with that, you just assumed that space is not a continuum, and there exists an indivisible base unit of it. (Which is probably true.) This heresy of course would get you burned at the stake at all official academical departments of math. :-)
I can't speak to the first and third examples, but the "painter's paradox" is silly, I think, not a paradox at all. It's easier to visualize in two dimensions: consider the function y = 1 / x ^ 2 (y equals one over x squared). We can start from 1 and head out to infinity, and the length of the line will be infinite, but the area under it will be finite. Extrude a fixed amount in and out of the page and we have an ugly three dimensional "horn" with the desired properties: finite volume and infinite surface area. But there's no paradox: you just have to keep painting in thinner and thinner coats the farther out you go. Of course, it soon becomes a tiny fraction of an atom's breadth in thickness, so no real paint would do, but there is no paradox.
You are right, but with that, you just assumed that space is not a continuum, and there exists an indivisible base unit of it. (Which is probably true.) This heresy of course would get you burned at the stake at all official academical departments of math. :-)
I love that the popular response to nonsense is "mind blown!"
Excellent piece. Funny, I *just* read about "Hilberts Hotel" 2 days ago in a book by Barrow about "nothing." Like minds lol