Infinity Paradox

Imagine a circle, for example. Say the circle's circumference is 2m. Take a 2m arc from it and you'll get the whole circle. Now take another circle with a 4m circumference; a 2m arc would be half of its circumference. With 8m, a 2m arc would be a fourth, and so on...



As we can see, as the size of the circle increases, a 2m arc from the circle loses its curvature, approaching the image of a straight line. If we put the earth beside the sun, it would appear that the sun has a flat surface, more so if we put the earth beside VY Canis Majoris, the largest star known as of now.

Let's now think of what would happen if the circle is of "infinite" size. What would the 2m arc look like? A straight line? Logically, yes if we apply basic calculus.

Here's another example. A regular polygon with three sides is an equilateral triangle; four sides, a square; five, a pentagon; et cetera. As the number of sides increases, the more it smooths out to become a round object. Set it to a polygon with infinite sides and it would become a circle, something that doesn't fit into the definition of a polygon.

But as common sense dictates, there is no such thing as a circle with a straight arc or a polygon with a round edge, is there? A circle can only be a single curved line and a polygon can only have straight sides. Moreover, it is hard to visualize "infinite". Visualizing a googol is hard enough, much less a googolplex... but infinite? Something with infinite attributes - size, power, intelligence, or age - just can't exist as far as reality is concerned.