[bytedude]

Flame wars over this used to be common on the internet. People intuitively have the notion that the left side approaches 1, but never actually equals it. They see it as a process instead of a fixed value. Maybe the notation is to blame.

[jhanschoo]

The intuition is right, and the mathematical definition relies on the intuition. It's just that people haven't been exposed to the actual definition when it comes to real numbers.

Mathematically, mathematicians prove that there is a unique number that this process goes to, (and not, say, two distinct numbers), and define the notation to represent this unique number.

[tgv]

Repetition can easily be seen as a process, which would indeed approach 1. But I think the idea of infinite repetition is very hard to get.

[username90]

The intuition that there is something in between isn't really wrong, it make sense and they work, otherwise physicists wouldn't be able to work with them. So that intuition is correct, it is mathematicians who just don't understand it fully yet. Maybe fully formalizing this is what unlocks the final piece keeping us from creating a unified theory in physics?

[steerablesafe]

Nonstandard analysis is a rigorous framework for working with infinitesimals (and infinitely large numbers).

https://en.wikipedia.org/wiki/Nonstandard_analysis