What is the value of $\aleph_1^{\aleph_0}$?

Question

Is there any neat way to calculate the value of $\aleph_1^{\aleph_0}$?

Answer 1

We have $2 \le \aleph_1 \le 2^{\aleph_0}$, so $2^{\aleph_0}\le \aleph_1^{\aleph_0}\le(2^{\aleph_0})^{\aleph_0}$. But $(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0 \cdot \aleph_0}=2^{\aleph_0}$, so $2^{\aleph_0}\le \aleph_1^{\aleph_0}\le2^{\aleph_0}$. Hence $\aleph_1^{\aleph_0}=2^{\aleph_0}$.

The same argument shows that $\lambda^\kappa=2^\kappa$ whenever $\aleph_0 \le \kappa$ and $2 \le \lambda \le 2^\kappa$.