The use of ‘infinity’ in a intersection/union of infinitely many sets. I understand the multiple meanings of infinity, per example, the difference between $\aleph_0$ and the $\infty$ in calculus limits as explained here: The Aleph numbers and infinity in calculus.
However, in the analysis book I am studying, in the chapter about the real number field, there is the standard question about the intersection of nested intervals (link1), which (unlike in link1) is presented as the intersection of infinitely many intervals: ${\cap}_1^{\infty}$ with $A_i$ being the closed intervals.
The chapter before this one was about set theory, but nowhere was this use of infinity really defined, nor the symbol itself, in an otherwise very rigorous and meticulous treatment of all other concepts. So my question is: what kind of infinity are we talking about here formally?
 A: Take a family of sets $\left\{A_i\right\}_{i\in I}$. We define
$$\bigcap_{i\in I}A_i=\left\{x:\forall i\in I\ x\in A_i\right\}$$
If you take the index set $I$ to be $\mathbb{N}$ we have 
$$\bigcap_{i\in \mathbb{N}}A_i$$
which is commonly denoted by
$$\bigcap_{i=0}^{\infty}A_i\text{ or }\bigcap_{i=1}^{\infty}A_i$$
depending on whether your definition of $\mathbb{N}$ contains $0$ or not. Therefore, your intersection is:
$$\bigcap_{i=1}^{\infty}A_i=\left\{x\in \mathbb{R}:\forall i\in \mathbb{N}\ x\in A_i\right\}$$
A: I would try not to think of the $\infty$ in $\cap_1^\infty$ as anything more than a symbol. The notation $\cap_1^\infty A_i$ (usually actually written $\cap_{i=1}^\infty A_i$) means the same thing as $\cap_{i\in \mathbb N} A_i$, that is the intersection of a collection of sets indexed by $\mathbb N$. If you really want to interpret $\infty$ as some infinity, since you are indexing over $\mathbb N$ (in a specific order) it should be interpreted as the ordinaltype of $\mathbb N$, which is $\omega_0$.
A: Usually the notation $\bigcap_{i=1}^\infty A_i$ would suggest that $i$ is an index running over the natural numbers from $1$ and it is unbounded, i.e. all the natural numbers (except $0$).
A: The $\infty$ that they're talking about should be: 
$\displaystyle\lim_{n\in \mathbb{N}} n $ , since the $A_i$'s are indexed by the natural numbers. 
