Does there exist a set containing infinite elements, whose elements themselves are sets containing infinite elements?

Does there exist a set containing infinite elements, whose elements themselves are sets containing infinite elements?

I think the answer is no, there is a famous paradox for it but I'm forgetting.

Solution 1:

There are many sets with this property. One example is $A=\{S\subseteq\mathbb{N}\mid |S|=\infty\}$, i.e. the set of infinite subsets of the naturals.

Solution 2:

There is no paradox here. Indeed, under the usual axioms of set theory (ZFC) there are lots of such sets.

What ZFC disallows is sets which contain themselves, and Russell's paradox (I suspect this might be what you're vaguely remembering) shows that we can't simultaneously have basic set formation axioms and a set of all sets. But there's no problem with infinite sets of infinite sets.

In fact, according to ZFC the "universe of sets" is built entirely from sets of sets of ...! Specifically, ZFC proves that every set $x$ occurs somewhere in the "tower" of sets $V_\alpha$, where

$V_0=\emptyset$
$V_{\beta+1}=\mathcal{P}(V_\beta)$ (here "$\mathcal{P}(X)$" is the powerset of $X$), and
$V_\alpha=\bigcup_{\beta<\alpha} V_\beta$ for $\alpha$ a limit.

Here $\alpha$ is an ordinal. If $\alpha$ is a finite ordinal, $V_\alpha$ will be finite; but once we go into the infinite ordinals we get all sorts of infinite sets, and infinite sets of infinite sets, and etc. So in fact this "sets-of-sets" stuff, which may feel paradoxical at first, is how ZFC interprets the entire mathematical universe!