×

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 \(\mathcal{M}_n\models \text{``Every homogeneously Suslin set of reals is `correctly-}\boldsymbol{\Delta}_{n+1}\)’.”

MSC:

03E15 Descriptive set theory
03E45 Inner models, including constructibility, ordinal definability, and core models
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] The higher infinite (2005)
[2] DOI: 10.1007/s00153-005-0301-3 · Zbl 1085.03040
[3] A minimal counterexample to universalbaireness 64 pp 1601– (1999)
[4] Woodin’s axiom (*), bounded forcing axioms, and precipitous ideals on {\(\omega\)}1 77 pp 475– (2012)
[5] Handbook of set theory 3 pp 1595– (2010)
[6] The core model iterability problem 8 (1996) · Zbl 0864.03035
[7] DOI: 10.1016/0168-0072(94)00021-T · Zbl 0821.03023
[8] Iterates of the core model 71 pp 241– (2006)
[9] Fine structure for tame inner models 61 pp 621– (1996)
[10] Handbook of set theory 3 pp 1877– (2010)
[11] Fine structure and iteration trees 3 (1994) · Zbl 0805.03042
[12] Descriptive set theory (1980) · Zbl 0433.03025
[13] DOI: 10.1090/S0894-0347-1989-0955605-X
[14] Journal of the American Mathematical Society 7 pp 1– (1989)
[15] The stationary tower: notes on a course by W. Hugh Woodin 32 (2004)
[16] DOI: 10.1016/0003-4843(72)90001-0 · Zbl 0257.02035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.