Humans have trouble imagining infinity. This trouble manifests itself in many different misunderstandings and debates in math, and this is one of them. You'll

Humans have trouble imagining infinity. This trouble manifests itself in many different misunderstandings and debates in math, and this is one of them. You'll see people say things like "0.999... approaches 1"; no, numbers don't "approach" things. Series can approach things, but they've substituted the number 0.999... with the infinite sum 9x10^-k for k = 1 -> inf (which it also is, and is also equal to 1), and then proceeded to misunderstand that, by imagining that series as being somehow built up over time. No, there's a qualitative, universe-shifting difference between "approaching infinity" and "at infinity", and the above series is defined to be "at infinity". People very often think they're talking about the latter when they're imagining the former. It's true that if you sat down and wrote a sequence of numbers with progressively more 9s at the end, you'd never reach 1. But that's because you never actually write the number we're talking about, which is equal to 1. Another place this shows up is in fractal dimensionality. If you imagine the Sierpinski triangle, it seems like nonsense to say that it is a 1.585-dimensional object. At every single iteration, it has a finite area: it's clearly 2-dimensional, right? But that's the error: you're not really imagining the Sierpinski triangle, because it's only defined at infinity; you're imagining it in steps approaching infinity. At every step approaching infinity, it is a 2-dimensional object with nonzero area. But at infinity, which is the only point it becomes a true fractal, it is a 1.585-dimensional object with no concept of 2-dimensional area.