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

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