Infinity in "Extended Natural Numbers" In Baby Rudin, p. 27, it is stated that the $\infty$ in notations like 
$\sum\limits_{i=0}^\infty i$
and 
$\bigcup\limits_{i=0}^\infty A_i$
is not the same as the $+\infty$ in the extended real number system (p. 11-12). 
This leads me to think that: we could create an extended natural number system, with $\infty$ as the supremum of every unbounded set in the standard natural number system. However, Rudin's statement would mean the $\infty$ in the extended natural numbers is not the same as the $+\infty$ of the real numbers. 
Why not?
 A: If you did in fact extend the natural numbers in that way, your $\infty$ would be the same as that of the extended real numbers. The real point here is that $\infty$ in the expressions $\sum_{i=0}^\infty a_i$ and $\bigcup_{i=0}^\infty A_i$ is actually not serving as a number of any kind: there is no term $a_\infty$ or set $A_\infty$ to be included in the sum or union. 
$\sum_{i=0}^\infty a_i$ is actually (in the calculus context) an abbreviation for a limit: it really means
$$\lim_{n\to\infty}\sum_{i=0}^na_i\;,$$
and the $\infty$ in that limit is again not a number: $\displaystyle\lim_{n\to\infty}$ is itself an abbreviation for a moderately complicated concept.
$\bigcup_{i=0}^\infty A_i$ is something different yet: it’s a conventional notation for what could better be written $$\bigcup_{i\in\Bbb N}A_i$$ or $$\bigcup\{A_i:i\in\Bbb N\}\;,$$ notations that are easily extended to unions over other index sets.
Added: Indeed, with unions and intersections there isn’t even an implicit notion of order involved, and we don’t actually need an index set. If $\mathscr{A}$ is any set of sets, we have
$$\bigcup\mathscr{A}=\left\{x:\exists A\in\mathscr{A}(x\in A)\right\}$$
and
$$\bigcap\mathscr{A}=\left\{x:\forall A\in\mathscr{A}(x\in A)\right\}\;.$$
