The tree property at the double successor
 of a measurable cardinal $\kappa $ with $2^{\kappa} $ large

Sy-David Friedman; Ajdin Halilović

doi:10.4064/fm223-1-4

Geodesic

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The tree property at the double successor of a measurable cardinal $\kappa $ with $2^{\kappa} $ large

Sy-David Friedman ¹ ; Ajdin Halilović ²

¹ Kurt Gödel Research Center University of Vienna 1090 Wien, Austria
² Faculty of Engineering Sciences Lumina&#8211;The University of South East Europe 021187 Bucureşti, Romania

Fundamenta Mathematicae, Tome 223 (2013) no. 1, pp. 55-64.

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

Résumé

Assuming the existence of a $\lambda ^+$-hypermeasurable cardinal $\kappa $, where $\lambda $ is the first weakly compact cardinal above $\kappa $, we prove that, in some forcing extension, $\kappa $ is still measurable, $\kappa ^{++}$ has the tree property and $2^\kappa =\kappa ^{+++}$. If the assumption is strengthened to the existence of a $\theta $-hypermeasurable cardinal (for an arbitrary cardinal $\theta >\lambda $ of cofinality greater than $\kappa $) then the proof can be generalized to get $2^\kappa =\theta $.

DOI : 10.4064/fm223-1-4

Keywords: assuming existence lambda hypermeasurable cardinal nbsp kappa where lambda first weakly compact cardinal above kappa prove forcing extension kappa still measurable kappa has tree property kappa kappa assumption strengthened existence theta hypermeasurable cardinal arbitrary cardinal theta lambda cofinality greater kappa proof generalized get kappa theta

Affiliations des auteurs :

Sy-David Friedman ¹ ; Ajdin Halilović ²

¹ Kurt Gödel Research Center University of Vienna 1090 Wien, Austria
² Faculty of Engineering Sciences Lumina&#8211;The University of South East Europe 021187 Bucureşti, Romania

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Sy-David Friedman; Ajdin Halilović. The tree property at the double successor
 of a measurable cardinal $\kappa $ with $2^{\kappa} $ large. Fundamenta Mathematicae, Tome 223 (2013) no. 1, pp. 55-64. doi : 10.4064/fm223-1-4. http://geodesic.mathdoc.fr/articles/10.4064/fm223-1-4/

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