= Solution
An <inner model> $M$ is <projectively well-ordered inner model>[projectively well-ordered] when some <projective set>[projective] relation well-orders the real numbers of $M$. The ordinal $\aleph_1^M=\omega_1^M$ is the least ordinal that $M$ regards as uncountable; equivalently, it is the supremum of the order types of the <well-order code>[well-order codes] in $M$.
Assume for contradiction that $\omega_1^M$ is uncountable in the ambient universe. Use the projective well-order of the reals of $M$ to choose, for each $\alpha<\omega_1^M$, the least $M$-real coding a well-order of type $\alpha$. Standard closure properties of the <projective hierarchy> make the resulting set $U\subseteq\mathrm{WF}$ projective. It is uncountable because it contains one distinct code for every $\alpha<\omega_1^M$.
The set $U$ has no perfect subset. Indeed, a perfect subset $P\subseteq U$ is closed and therefore <analytic set>[analytic]. The <boundedness theorem for well-order codes> bounds the ranks of its members below one countable ordinal $\beta$. Since $U$ contains at most one code of each rank, $P$ would then be countable, whereas every nonempty perfect set of reals is uncountable.
If every projective set is determined, <projective determinacy> holds and gives the <perfect set property> to every projective set. Applying it to the uncountable projective set $U$ yields a perfect subset, a contradiction. Therefore
$$
\boxed{\omega_1^M\text{ is countable in the ambient universe}.}
$$
This is <projective determinacy collapses the inner-model omega-one>.
Back to article page
