Okay, so I'm laying in bed all night unable to sleep. And I start getting a weird idea, which makes me laugh. But as I'm in my stupor pondering the matter I concluded that there's enough there to be worth a post to check.

So. Limits. $\displaystyle \lim_{x \to \infty}$. Which infinity?

Hear me out. I realize if the limit were done over a variable x taken to an $\displaystyle \aleph _2$ infinity we might just embed the space we want to work in into a suitably large Euclidean n-space and just do the limit there without need for any funky theorems.

But I digress.

I've always assumed the "infinity" in the limit statement is $\displaystyle \aleph _1$ and not something "larger." But might there be a reason we would consider the limit to be taken to $\displaystyle \aleph _2$? Some of the rationality behind this is based on the principle of transfinite induction, where the index set can be uncountably infinite. I realize there's no direct relationship between the two ideas, but I've played around with transfinite induction (for a whole half-hour!) and thought "Why not ask? I'm a Physics nerd with overtones of a Mathematical superiority complex and it's a Friday night!"

-Dan