Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 121 / 1 / v / c

The collapse $\pi :H\to N$ is a bijection, so $|N|=|H|={\aleph }_1$. Every $\alpha <{\omega }_1$ and every element of $\alpha$ belong to $H$. The transitive collapse fixes transitive subsets lemma therefore gives $\pi (\alpha )=\alpha$, so every ambient countable ordinal belongs to $N$.

Moreover, $V_{\kappa }$ correctly regards each such $\alpha$ as countable. An injection $\alpha \to \omega$ has rank below $\max (\alpha ,\omega )+\omega <\kappa$, so a witness belongs to $V_{\kappa }$. Elementarity gives $H\models$ “$\alpha$ is a countable ordinal”. Transporting this statement across the collapse, which fixes $\alpha$, gives the same statement in $N$.

The collapse also fixes the element ${\omega }_1\in H$. No injection ${\omega }_1\to \omega$ exists externally, hence none belongs to $V_{\kappa }$; together with the countability witnesses below it, this shows that $V_{\kappa }$ correctly identifies ${\omega }_1$ as its first uncountable ordinal. Elementarity and the collapse preserve that identification. Thus the construction gives