Simple example for a set with cardinality of $\aleph_2$ What is a simple or commonly known set that has a cardinality equal to $\aleph_2$ or greater?
 A: The problem with alephs, other than $\aleph_0$, is that they don't really live in the 'commonly known' world. Think of the sets that are commonly known by mathematicians:


*

*Finite sets;

*Number sets $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$, which have cardinality $\aleph_0$ or $2^{\aleph_0}$;

*Power sets, which have cardinalities of the form $2^{\kappa}$;

*Function sets, which have cardinalities of the form $\kappa^{\lambda}$;

*Sets of sequences (see function sets);

*Products and unions of sets, which have cardinalities of the form $\kappa\lambda$ or $\kappa + \lambda$;

*...and so on.


The realm of the 'commonly known' consists of cardinals formed from finite cardinals and $\aleph_0$ by taking powers, sums and products, not cardinal successors.
I'd say the most commonly known set of cardinality equal to $\aleph_2$ is the (von Neumann) ordinal $\omega_2$, or indeed any ordinal $\alpha$ for which $\omega_2 \le \alpha < \omega_3$.
However, we can certainly construct cardinals which are greater than or equal to $\aleph_2$. Indeed, we know that $2^{\kappa} > \kappa$ for all cardinals $\kappa$, so
$$2^{2^{\aleph_0}} > 2^{\aleph_0} > \aleph_0 \quad \Rightarrow \quad 2^{2^{\aleph_0}} \ge (2^{\aleph_0})^+ \ge \aleph_0^{++} = \aleph_2$$
and hence any set of cardinality $2^{2^{\aleph_0}}$ will do for you. For instance $\mathcal{P}(\mathbb{R})$ or $\mathbb{R}^{\mathbb{R}}$, both of which are sets appearing in (say) real analysis.
A: Sets of ℵ2 cardinality are quite uncommon but it seem we have some example with real closed fields : under continuum hypothesis, all real closed fields of ℵ1 cardinality that also satisfy η1 property (that is : for two subsets A and B  of cardinality less than ℵ1, such as any element of A is less than any element of B, there is an element of X greater than any element of A and less than any element of B) are isomorphic to the hyperreal field F, which is apparently not complete, that is it is included and dense in an ordered field K, and its completion K is of cardinality ℵ2. 
