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STABLY MEASURABLE CARDINALS

Part of: Set theory

Published online by Cambridge University Press:  15 June 2020

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 define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma _{1}$ -definability at uncountable regular cardinals. In particular we give its exact consistency strength first in terms of the second uniform indiscernible for bounded subsets of $\kappa $ : $u_2(\kappa )$ , and secondly to give the consistency strength of a property of Lücke’s.

TheoremThe following are equiconsistent:

  1. (i) There exists $\kappa $ which is stably measurable;

  2. (ii) for some cardinal $\kappa $ , $u_2(\kappa )=\sigma (\kappa )$ ;

  3. (iii) The $\boldsymbol {\Sigma }_{1}$ -club property holds at a cardinal $\kappa $ .

Here $\sigma (\kappa )$ is the height of the smallest $M \prec _{\Sigma _{1}} H ( \kappa ^{+} )$ containing $\kappa +1$ and all of $H ( \kappa )$ . Let $\Phi (\kappa )$ be the assertion:

TheoremAssume $\kappa $ is stably measurable. Then $\Phi (\kappa )$ .

And a form of converse:

TheoremSuppose there is no sharp for an inner model with a strong cardinal. Then in the core model K we have: $\mbox {``}\exists \kappa \Phi (\kappa ) \mbox {''}$ is (set)-generically absolute ${\,\longleftrightarrow \,}$ There are arbitrarily large stably measurable cardinals.

When $u_2(\kappa ) < \sigma (\kappa )$ we give some results on inner model reflection.

Keywords

inner modelstabilitylarge cardinalsreflection

MSC classification

Primary: 03E35: Consistency and independence results
Secondary: 03E45: Inner models, including constructibility, ordinal definability, and core models 03E55: Large cardinals
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 2 , June 2021 , pp. 448 - 470
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Copyright
© The Association for Symbolic Logic 2020

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References

Aczel, P. and Richter, W., Inductive definitions and reflecting properties of admissible ordinals , Generalised Recursion Theory, Studies in Logic and the Foundation of Mathematics, vol. 79, North-Holland, Amsterdam, 1974, pp. 301–381.Google Scholar
Barton, N., Caicedo, A., Fuchs, G., Hamkins, J. D., Reitz, J., and Schindler, R.-D., Inner model reflection principles . Studia Logica , vol. 108 (2019), p. 578.Google Scholar
Dodd, A. J., The Core Model , London Mathematical Society Lecture Notes in Mathematics, vol. 61, Cambridge University Press, Cambridge, 1982.CrossRefGoogle Scholar
Donder, H.-D., Jensen, R. B., and Koppelberg, B., Some applications of K , Set Theory and Model Theory (Jensen, R. and Prestel, A., editors), Springer Lecture Notes in Mathematics, vol. 872, Springer Verlag, New York, 1981, pp. 55–97.CrossRefGoogle Scholar
Feng, Q., Magidor, M., and Woodin, W. H., Universally Baire sets of reals , Set Theory of the Continuum (Judah, H., Just, W., and Woodin, W. H., editors), MSRI Publications, Springer Verlag, New York, 1992.Google Scholar
Jensen, R. B., The fine structure of the constructible hierarchy . Annals of Mathematical Logic , vol. 4 (1972), pp. 229–308.CrossRefGoogle Scholar
Kanamori, A., The Higher Infinite , Springer Monographs in Mathematics, second ed., Springer Verlag, New York, 2003.Google Scholar
Lücke, P., Partition properties for simply definable colourings, 2018, arXiv.Google Scholar
Lücke, P., Schindler, R.-D., and Schlicht, P., ${\varSigma}_1\left(\kappa \right)$ -definable subsets of $H\left({\kappa}^{+}\right)$ , this Journal , vol. 82 (2017), no. 3, pp. 1106–1131.Google Scholar
Lücke, P. and Schlicht, P., Measurable cardinals and good ${\varSigma}_1\left(\kappa \right)$ -wellorderings of $H\left({\kappa}^{+}\right)$ . Mathematical Logic Quarterly , vol. 64 (2018), p. 207.CrossRefGoogle Scholar
Mitchell, W. J., Ramsey cardinals and constructibility, this Journal, vol. 44 (1979), no. 2, pp. 260–266.Google Scholar
Sharpe, I. and Welch, P. D., Greatly Erdős cardinals and some generalizations to the Chang and Ramsey properties . Annals of Pure and Applied Logic , vol. 162 (2011), pp. 863–902.CrossRefGoogle Scholar
Welch, P. D., Doing without determinacy – aspects of inner models , Proceedings of the Logic Colloquium Hull ‘86, Series in Logic and its Applications (Drake, F. and Truss, J., editors), North-Holland Publishing Co., Amsterdam, 1988, pp. 333–342.Google Scholar
Welch, P. D., Some descriptive set theory and core models . Annals of Pure and Applied Logic , vol. 39 (1988), pp. 273–290.CrossRefGoogle Scholar
Welch, P. D., Characterising subsets of ${\omega}_1$ constructible from a real, this Journal, vol. 59 (1994), no. 4, pp. 1420–1432.Google Scholar
Welch, P. D., On unfoldable cardinals, $\omega$ -closed cardinals, and the beginnings of the inner model hierarchy . Archive for Mathematical Logic , vol. 43 (2004), no. 4, pp. 443–458.10.1007/s00153-003-0199-6CrossRefGoogle Scholar
Zeman, M., Inner Models and Large Cardinals , Series in Logic and its Applications, vol. 5, de Gruyter, Berlin, New York, 2002.10.1515/9783110857818CrossRefGoogle Scholar