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

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.