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Forcing indestructibility of set-theoretic axioms

Published online by Cambridge University Press:  12 March 2014

Bernhard KÖnig
Bernhard KÖnig*
Affiliation:
UniversitÉ Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada. E-mail: [email protected]
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Abstract

Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Lévy collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to ℵ1. Later we give applications, among them the consistency of MM with ℵω not being Jónsson which answers a question raised in the set theory meeting at Oberwolfach in 2005.

Keywords

Forcing axiomstransfer principles
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 1 , March 2007 , pp. 349 - 360
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Bagaria, Joan, Bounded forcing axioms as principles of generic absoluteness, Archive for Mathematical Logic, vol. 39 (2000), pp. 393–401.CrossRefGoogle Scholar
[2]Baumgartner, James, Applications of the Proper Forcing Axiom, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J.E., editors), North-Holland, 1984, pp. 913–959.Google Scholar
[3]Cummings, James, Large cardinal properties of small cardinals, Set theory (Curacao, 1995; Barcelona, 1996), Kluwer Academic Publishers, 1998, pp. 23–39.Google Scholar
[4]Cummings, James, Foreman, Matthew, and Magidor, Menachem, Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), pp. 35–98.CrossRefGoogle Scholar
[5]Foreman, Matthew, Games played on Boolean algebras, this Journal, vol. 48 (1983), pp. 714–723.Google Scholar
[6]Foreman, Matthew, Magidor, Menachem, and Shelah, Saharon, Martin's Maximum, saturated ideals, and nonregular ultrafilters I, Annals of Mathematics, vol. 127 (1988), pp. 1–47.Google Scholar
[7]Kanamori, Akihiro, The Higher Infinite, Perspectives in Mathematical Logic, Springer-Verlag, 1997.CrossRefGoogle Scholar
[8]Bernhard, König, Generic compactness reformulated, Archive for Mathematical Logic, vol. 43 (2004), pp. 311–326.Google Scholar
[9]König, Bernhard and Yoshinobu, Yasuo, Fragments of Martin's Maximum in generic extensions, Mathematical Logic Quarterly, vol. 50 (2004), pp. 297–302.CrossRefGoogle Scholar
[10]König, Bernhard and Yoshinobu, Yasuo, Kurepa-trees and Namba forcing, preprint, 2005.Google Scholar
[11]Kunen, Kenneth, Saturated ideals, this Journal, vol. 43 (1978), pp. 65–76.Google Scholar
[12]Kunen, Kenneth, Set theory, an introduction to independence proofs, North-Holland, 1980.Google Scholar
[13]Larson, Paul, Separating stationary reflection principles, this Journal, vol. 65 (2000), pp. 247–258.Google Scholar
[14]Moore, Justin, Set mapping reflection, Journal of Mathematical Logic, vol. 5 (2005), pp. 87–97.Google Scholar
[15]Shelah, Saharon, On successors of singulars. Logic colloquium '78, Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, 1979, pp. 357–380.Google Scholar
[16]Shelah, Saharon, Proper forcing, Lecture notes in Mathematics, vol. 940, Springer-Verlag, 1982.CrossRefGoogle Scholar
[17]Shelah, Saharon, Proper and Improper Forcing, Perspectives in Mathematical Logic, Springer-Verlag, 1998.CrossRefGoogle Scholar
[18]Todorcevic, Stevo, Localized reflection and fragments of PFA, Dimacs series, vol. 58, 2002, pp. 135–148.Google Scholar
[19]Velickovic, Boban, Forcing axioms and stationary sets, Advances in Mathematics, vol. 94 (1992), pp. 256–284.CrossRefGoogle Scholar
[20]Woodin, W. Hugh, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter & Co., Berlin, 1999.CrossRefGoogle Scholar
[21]Yoshinobu, Yasuo, Approachability and games onposets, this Journal, vol. 68 (2003), pp. 589–606.Google Scholar