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 $\aleph_2$.

 

[tacman]

Yes, there are several examples of sets with cardinality \aleph_{2}. One example is the set of all countable ordinals, denoted by \omega_{2}. This set has cardinality \aleph_{2} because it contains all countable ordinals up to and including the second uncountable ordinal, \omega_{2}. Another example is the set of all subsets of the real numbers with cardinality \aleph_{1}, denoted by \mathcal{P}(\mathbb{R})_{\aleph_{1}}. This set has cardinality \aleph_{2} because there are \aleph_{1} subsets of \mathbb{R}, and each of these subsets has cardinality \aleph_{1}, resulting in a total cardinality of \aleph_{1} \cdot \aleph_{1} = \aleph_{2}. Additionally, the set of all functions from \mathbb{R} to \mathbb{R} has cardinality \aleph_{2} because there are \aleph_{2} real numbers and each function can be represented as a subset of \mathbb{R} \times \mathbb{R}, resulting in a total cardinality of \mathcal{P}(\mathbb{R} \times \mathbb{R}) = \aleph_{2}. These are just a few examples, as there are many other sets with cardinality \aleph_{2} in higher cardinality mathematics.