Higher Cardinals: Example of Set with Cardinality Aleph2?

[Son Goku]

Just an ideal question, possibly asked before, but is there an example of a Set which has cardinality of $$\aleph_{2}$$?

 

[CRGreathouse]

That's a tough one, since it depends on the assumptions you use. With the GCH, the set of all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ has cardinality $\aleph_2$.