The consistency strength of the tree property at the double successor of a measurable cardina

Natasha Dobrinen¹ ; Sy-David Friedman²

¹ Department of Mathematics University of Denver 2360 S Gaylord St Denver, CO 80208, U.S.A.
² Kurt Gödel Research Center for Mathematical Logic Universität Wien Währinger Strasse 25 A-1090 Wien, Austria

Fundamenta Mathematicae, Tome 208 (2010) no. 2, pp. 123-153

Résumé

The Main Theorem is the equiconsistency of the following two statements: (1) $\kappa$ is a measurable cardinal and the tree property holds at $\kappa^{++}$; (2) $\kappa$ is a weakly compact hypermeasurable cardinal.From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the double successor of each strongly inaccessible cardinal. If $0^{\#}$ exists, then we can construct an inner model in which the tree property holds at the double successor of each strongly inaccessible cardinal. We also find upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double successor of a measurable cardinal. The upper and lower bounds differ only by 1 in the Mitchell order.

Détail

DOI : 10.4064/fm208-2-2

Keywords: main theorem equiconsistency following statements kappa measurable cardinal tree property holds kappa kappa weakly compact hypermeasurable cardinal proof main theorem internal consistency results follow there weakly compact hypermeasurable cardinal measurable cardinal far enough above there inner model which there proper class measurable cardinals which tree property holds double successor each strongly inaccessible cardinal exists construct inner model which tree property holds double successor each strongly inaccessible cardinal upper lower bounds consistency strength there being special aronszajn trees double successor measurable cardinal upper lower bounds differ only mitchell order