Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 24 / 6 / b / ii

The PDF's middle equality is $({\omega }^2{)}^M=({\omega }^2{)}^{M[G]}$, concerning ordinal exponentiation. The TeX incorrectly turns the exponent into a subscript; preserving ${\omega }_2$ would be incompatible with forcing CH over many ground models.

Let $A=(\mathcal{P}(\omega ){)}^M$ and use the countable-condition collapse of $A$ onto ${\omega }_1$:ordered by extension. Countable unions give lower bounds, so closed forcing adds no short ordinal sequences; in particular it adds no reals and preserves ${\omega }_1$. Dense requirements ensure that the union of the generic filter is a total surjection $g:{\omega }_1\to A$. ThusFinally forcing preserves ordinals, and the recursion defining ordinal multiplication and exponentiation is absolute. Both models compute ${\omega }^2$ as the same order type of $\omega$ successive copies of $\omega$. This proves the printed middle equality too. Higher cardinals may collapse, which is allowed by the actual PDF.