Exponential growth does mean that there exists some interval length over which the function doubles regardless of your point in the curve.

It's certainly true that it won't be true for all interval lengths - that is obviously mathematically nonsensical.

> that is obviously mathematically nonsensical.

In a finite universe, yes, it is clearly nonsense. Exponential growth cannot continue forever in a finite universe, no matter how small the growth constant.

Your comment is unrelated to what I was discussing in the quoted section.

I was saying you can't have growth for which f(n + k) = 2 f(n) for all k > 0 (I believe the only solution to that is f(n) = 0, which isn't what anyone is talking about when they talk about "exponential growth"...). This point was intended, in contrast with what I'd just said about there existing some k - to stress the notion of when the interval is picked.

I know. I was agreeing with you. In fact, I was giving a proof that your position was correct.

Apparently, I wasn't doing it well, but that's what I was trying to do.

I didn't think you were disagreeing with me - I think you were agreeing with something I didn't say. I was just speaking to definitions.

Whether unbounded exponential "economic growth" is possible in a finite universe is actually a significantly more involved question. It's obviously not possible if "economic value" is bounded, but in principle could be "possible" with finite stuff in continuous space (obviously, economic value depends significantly on configurations of stuff and not just amounts).

All of this is interesting, but I honestly don't expect any reasoning involving hard theoretic bounds to be of any use in determining what's likely over my lifetime, or those of my children.