Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2010 / B / 5 / d

Constant functions give the lower bound ${\aleph }_2\le {\aleph }_2^{{\aleph }_0}$. By the Generalized continuum hypothesis, ${\aleph }_2=2^{{\aleph }_1}$. The currying law for cardinal exponentiation and infinite cardinal arithmetic therefore giveHence $\boxed{{\aleph }_2^{{\aleph }_0}={\aleph }_2}$.

Put $\kappa ={\aleph }_{\omega }$. Apply König theorem for cardinal numbers to the sequences ${\aleph }_n<{\aleph }_{n+1}$:The sum is $\kappa$ because the countable union has cardinality at most $\kappa$ and its summands are cofinal in $\kappa$. Thus the power is strictly larger than $\kappa$, a fact that cannot be replaced by the regular-cardinal calculation of the preceding part.

For an upper bound, use $\kappa \le 2^{\kappa }$ and the currying law for cardinal exponentiation:where the last step is GCH at $\kappa$. There is no cardinal strictly between $\kappa$ and its successor, so

For $\kappa ={\aleph }_{\omega }$, monotonicity of cardinal exponentiation givesThe middle equality uses the currying law for cardinal exponentiation, and the final one uses ${\aleph }_0\cdot \kappa =\kappa$. Hence ${\aleph }_0^{\kappa }=2^{\kappa }$, even before assuming GCH. Applying the Generalized continuum hypothesis yields