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CLOSED AND UNBOUNDED CLASSES AND THE HÄRTIG QUANTIFIER MODEL

Part of: Set theory

Published online by Cambridge University Press:  15 February 2021

PHILIP D. WELCH
PHILIP D. WELCH*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF BRISTOLBRISTOLBS8 1TW, UKE-mail:p.welch@bristol.ac.uk
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Abstract

We show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P, Q, {\langle L[P],\in ,P \rangle }$ and ${\langle L[Q],\in ,Q \rangle }$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. Examples of such P are Card, the class of uncountable cardinals, I the uniform indiscernibles, or for any n the class $C^{n}{=_{{\operatorname {df}}}}\{ \lambda \, | \, V_{\lambda } \prec _{{\Sigma }_{n}}V\}$ ; moreover the theory of such models is invariant under ZFC-preserving extensions. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. The inner model constructed using definability in the language augmented by the Härtig quantifier is thus also characterized.

Keywords

inner modelclosed and unbounded classPrikry forcing

MSC classification

Primary: 03E10: Ordinal and cardinal numbers
Secondary: 03E45: Inner models, including constructibility, ordinal definability, and core models 03E55: Large cardinals
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 2 , June 2022 , pp. 564 - 584
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Dodd, A. J., The Core Model , London Mathematical Society Lecture Notes in Mathematics, vol. 61, Cambridge University Press, Cambridge, 1982.10.1017/CBO9780511600586CrossRefGoogle Scholar
Fuchs, G., A characterisation of generalized Příkrý forcing . Archive for Mathematical Logic , vol. 44 (2005), pp. 935–971.10.1007/s00153-005-0313-zCrossRefGoogle Scholar
Gitik, M., Prikry type forcings , Handbook of Set Theory, vol. 2 (Foreman, M. and Kanamori, A., editors), Springer, New York, 2007, pp. 1351–1447.Google Scholar
Herre, H., Krynicki, M., Pinus, A., and Väänänen, J., The Härtig quantifier: A survey , This Journal , vol. 56 (1991), no. 4, pp. 1153–1183.Google Scholar
Jensen, R. B., The Core Model for Measures of Order Zero, unpublished circulated manuscript , Oxford, 1989.Google Scholar
Kanamori, A., The Higher Infinite , Omega Series in Logic, Springer, New York, 1994.Google Scholar
Kennedy, J., Magidor, M., and Väänänen, J., Inner Models from Extended Logics , Isaac Newton Preprint Series, vol. NI 16006, Cambridge, 2016.Google Scholar
Koepke, P., The consistency strength of the free subset problem for ωω , This Journal , vol. 49 (1984), no. 4, pp. 1198–1204.Google Scholar
Magidor, M., How large is the first strongly compact cardinal? Or a study in identity crises . Annals of Mathematical Logic , vol. 10 (1974), pp. 33–57.10.1016/0003-4843(76)90024-3CrossRefGoogle Scholar
Steel, J. R., Determinacy in the Mitchell models . Annals of Mathematical Logic , vol. 22 (1980), pp. 109–125.10.1016/0003-4843(82)90017-1CrossRefGoogle Scholar
Steel, J. R. and Welch, P. D., ${\varSigma}_3^1$ -absoluteness and the second uniform indiscernible. Israel Journal of Mathematics , vol. 104 (1998), pp. 157–190.10.1007/BF02897063CrossRefGoogle Scholar
Welch, P. D., Determinacy in the difference hierarchy of co-analytic sets . Annals of Pure and Applied Logic , vol. 80 (1996), pp. 69–108.10.1016/0168-0072(95)00057-7CrossRefGoogle Scholar
Woodin, W. H., The universe constructed from a sequence of ordinals . Archive for Mathematical Logic , vol. 35 (1996), pp. 371–383.10.1007/s001530050051CrossRefGoogle Scholar
Zeman, M., Inner Models and Large Cardinals , Series in Logic and Its Applications, vol. 5, De Gruyter, Berlin and New York, 2002.10.1515/9783110857818CrossRefGoogle Scholar