Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 158 / 3

An inner model $M$ is projectively well-ordered when some 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 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. 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. ThereforeThis is projective determinacy collapses the inner-model omega-one.

Suppose ZFC proved that every ${\aleph }_1$-Suslin set is determined. The assumed consistency of ZFC and the relative consistency of the Continuum hypothesis would then give a model ofIn that model $2^{{\aleph }_0}={\aleph }_1$. By Every set of reals is continuum-Suslin, every subset of ${\omega }^{\omega }$ is therefore ${\aleph }_1$-Suslin and hence determined. This is the axiom of determinacy.

But the axiom of choice produces an undetermined set of reals, so ZFC and the axiom of determinacy are incompatible. The displayed theory cannot have a model, contradicting the relative consistency of ZFC plus the continuum hypothesis. Hence ZFC cannot prove that all ${\aleph }_1$-Suslin sets are determined.