Homogeneously Suslin sets in tame mice. (English) Zbl 1282.03022

D. A. Martin (1968) proved that if a set of reals is \(\kappa\)-homogeneously Suslin, then it is determined. Martin also proved that if \(\kappa\) is a measurable cardinal, then every (boldface) \(\boldsymbol{\Pi}_1^1\) set of reals is \(\kappa\)-homogeneously Suslin. J. R. Steel has shown that \(L[\mu]\models\) “A set of reals \(A\) is homogeneously Suslin if and only if \(A\) is \(\boldsymbol{\Pi}_1^1\).”
Homogeneously Suslin sets continue to be prominent in the study of determinacy hypotheses and in inner model theory. In the paper under review, the author shows that the above equivalence also holds in a particular active mouse denoted by \(0^{¶}\) (the sharp of a strong cardinal). Other related results are also established. For each \(n\in\omega\), let \(\mathcal{M}_n\) be the canonical inner model satisfying “there exist \(n\) Woodin cardinals.” Martin, Mitchell, and Steel have shown that the inner model \(\mathcal{M}_n\) is \(\Sigma_{n+1}^1\)-correct, that is, \(\mathbb{R}^{\mathcal{M}_n}\preccurlyeq_{\Sigma_{n+1}^1} \mathbb{R}^{V}\). Let \(z\) be a real in \(\mathcal{M}_n\) and let \(A\) be a set of reals in \(\mathcal{M}_n\). Then \(A\) is said to be \(\mathcal{M}_n\)-correctly-\(\Delta_{n+1}^1(z)\) if and only if there is a \(\Delta_{n+1}^1(z)\) set \(B\) in \(V\) such that \(A=B\cap \mathcal{M}_n\). The author notes that the class of \(\mathcal{M}_n\)-correctly-\(\boldsymbol{\Delta}_{n+1}\) sets of reals can be defined, indirectly, within \(\mathcal{M}_n\). The author then shows that \(\mathcal{M}_n\models \text{``Every homogeneously Suslin set of reals is `correctly-}\boldsymbol{\Delta}_{n+1}\)’.”


03E15 Descriptive set theory
03E45 Inner models, including constructibility, ordinal definability, and core models
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