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PFA AND THE DEFINABILITY OF THE NONSTATIONARY IDEAL

Part of: Set theory

Published online by Cambridge University Press:  11 April 2025

STEFAN HOFFELNER Open the ORCID record for STEFAN HOFFELNER [Opens in a new window] ,
PAUL LARSON Open the ORCID record for PAUL LARSON [Opens in a new window] ,
RALF SCHINDLER Open the ORCID record for RALF SCHINDLER [Opens in a new window]  and
LIUZHEN WU Open the ORCID record for LIUZHEN WU [Opens in a new window]
STEFAN HOFFELNER
Affiliation:
INSTITUT FÜR MATHEMATISCHE LOGIK UNIVERSITÄT MÜNSTER EINSTEINSTR. 62, MÜNSTER GERMANY E-mail: stefan.hoffelner@gmx.at E-mail: rds@uni-muenster.de
PAUL LARSON*
Affiliation:
DEPARTMENT OF MATHEMATICS MIAMI UNIVERSITY OXFORD, OHIO 45056
RALF SCHINDLER
Affiliation:
INSTITUT FÜR MATHEMATISCHE LOGIK UNIVERSITÄT MÜNSTER EINSTEINSTR. 62, MÜNSTER GERMANY E-mail: stefan.hoffelner@gmx.at E-mail: rds@uni-muenster.de
LIUZHEN WU
Affiliation:
INSTITUTE OF MATHEMATICS ACADEMY OF MATHEMATICS AND SYSTEMS SCIENCE CHINESE ACADEMY OF SCIENCES BEIJING 100190 CHINA and SCHOOL OF MATHEMATICAL SCIENCES UNIVERSITY OF CHINESE ACADEMY OF SCIENCES BEIJING 100049 CHINA E-mail: lzwu@math.ac.cn
*
E-mail: larsonpb@miamioh.edu
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Abstract

We produce, relative to a $\textsf {ZFC}$ model with a supercompact cardinal, a $\textsf {ZFC}$ model of the Proper Forcing Axiom in which the nonstationary ideal on $\omega _1$ is $\Pi _1$-definable in a parameter from $H_{\aleph _2}$.

Keywords

proper forcing axiomnonstationary idealforcingcountable support iteration

MSC classification

Primary: 03E35: Consistency and independence results 03E47: Other notions of set-theoretic definability

Information

Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 8
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

REFERENCES

Foreman, M., Magidor, M., and Shelah, S., Martin’s maximum, saturated ideals, and non-regular ultrafilters. Part I . Annals of Mathematics, vol. 127 (1988), no. 1, pp. 1–47.Google Scholar
Fuchs, U., Donder’s version of revised countable support, preprint, 2008. arxiv.org/pdf/math/9207204v1.pdf.Google Scholar
Hoffelner, S., Larson, P., Schindler, R., and Wu, L., Forcing axioms and the definability of  $\kern1.2pt{\sf NS}_{\omega_1}$ . Journal of Symbolic Logic, vol. 89 (2024), no. 4, pp. 1641–1658.Google Scholar
Jech, T., Set Theory, Springer-Verlag, Berlin, 2003.Google Scholar
Kunen, K., Set Theory: An Introduction to Independence Proofs, North Holland, Amsterdam-New York, 1980.Google Scholar
Larson, P., The size of $\overset{\sim }{T}$ . Archive for Mathematical Logic, vol. 39 (2000), pp. 541–568.Google Scholar
Larson, P., The Stationary Tower, University Lecture Series, 32, American Mathematical Society, Providence, RI, 2004.Google Scholar
Laver, R., Making the supercompactness of $\kappa$ indestructible under $\kappa$ -directed closed forcing . Israel Journal of Mathematics, vol. 29 (1978), no. 4, pp. 385–388.Google Scholar
Lücke, P., Schlicht, P., and Schindler, R., Lightface definable subsets of ${H}_{\omega_2}$ . The Journal of Symbolic Logic, vol. 82 (Sep. 2017), no. 3, pp. 1106–1131.Google Scholar
Miyamoto, T., A limit stage construction for iterating semiproper preorders , Proceedings of the 7th and 8th Asian Logic Conferences, Singapore University Press, Singapore, 2003, pp. 303–327.Google Scholar
Schindler, R. and Sun, X., in preparation.Google Scholar
Shelah, S., Proper and Improper Forcing, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.Google Scholar
Woodin, W. H., The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, de Gruyter, Berlin, 1999.Google Scholar