Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 16H / e / Solution

Apply König theorem for cardinal numbers with $I=\omega$, ${\kappa }_i={\aleph }_i$, and ${\lambda }_i={\aleph }_{i+1}$. Standard infinite cardinal arithmetic giveswhereas every factor in the product is at most ${\aleph }_{\omega }$. Hence

Now suppose $2^{{\aleph }_{\omega }}={\aleph }_{\omega }$. The exponent laws for infinite cardinals would givecontradicting the strict inequality. Therefore