 Infinity lies at the heart of a great deal of confusion and error in the world of mathematics, modern mathematics in particular. Mathematicians think that they can do things that they actually can't do. They think they can conceive of things that they actually can't conceive of. Let me give you an excellent example. A circle whose radius is infinite in length. Ooh, an infinite circle. Can you make sense of such a concept? Mathematicians might say, yes, of course, we can do all kinds of mathematics with this idea of a circle with the radius is infinite in length. Logicians and philosophers, maybe some more careful thinkers, might say, actually that doesn't make any sense. In fact, when you throw infinity into the mix, things start getting all wonky. One question illustrates the problem. Take any line segment on that infinite circle and answer this question. What is the curvature of that line segment? There are two possible answers. It is either the case that there is a curvature or there is no curvature. So if it's true that this line segment has a curvature, a non-zero curvature, whatever it is, then we are necessarily logically talking about a finite circle. You follow the curvature all the way around and you've constructed yourself a circle. It literally doesn't matter how big the circle is. It doesn't matter how small or just tiny fraction of a curvature. If there is any curvature whatsoever, you are necessarily talking about a finite circle. But this presents us with a difficult problem. If there is no curvature, then that means we're literally talking about a straight line. Yes, so that means, in fact, a circle with a radius that is infinite in length is identical to a straight line. And in fact, as far out as you go along that straight line, there is never any curvature. Because if there were any curvature, it would necessarily be a finite circle. Does that strike you as odd? A circle that has no curvature logically cannot have any curvature. I suggest if you're talking about a circle that has absolutely zero curvature, then you're talking about a line and you're obviously not talking about a circle. Conceptually, this is, I would say, explicitly logically contradictory. There is no such thing as a line, which is a circle. Now, I'm talking about Euclidean space here. I might all talk about non-Euclidean space in another video. But these concepts are logically, mutually exclusive, and yet mathematicians don't treat them as such. This is just one counterintuitive example of the nature of infinity. Modern mathematics is shot throughout with counterintuitive conclusions like that, where we're talking about infinite sets. Say the concept of a set is necessarily a collection, a discrete collection of elements. They say, no, no, there are these things as infinite sets. That they're collections, but there is a never-ending amount of them. There isn't actually a finite amount, and yet we can collect all of them together into one discrete thing. If that strikes you as logically contradictory for the same reasons that there are no infinite circles, then good for you, you have a good head on your shoulders. However, infinite sets are stuck at the very root of modern mathematics. Set theory is considered one of the foundational areas of modern math, and it contains actualized infinities shot throughout. Eventually, I'm going to cover more of these logical paradoxes that are involved with infinity, but suffice to say, everywhere that infinity rears its head, when it's actualized, when it's completed, this idea of a completed infinity causes logical errors.