The tree property at the successor of a singular limit of measurable cardinals

Abstract

Assume $$\lambda $$ is a singular limit of $$\eta $$ supercompact cardinals, where $$\eta \le \lambda $$ is a limit ordinal. We present two methods for arranging the tree property to hold at $$\lambda ^{+}$$ while making $$\lambda ^{+}$$ the successor of the limit of the first $$\eta $$ measurable cardinals. The first method is then used to get, from the same assumptions, the tree property at $$\aleph _{\eta ^2+1}$$ with the failure of SCH at $$\aleph _{\eta ^2}$$ . This extends results of Neeman and Sinapova. The second method is also used to get the tree property at the successor of an arbitrary singular cardinal, which extends some results of Magidor–Shelah, Neeman and Sinapova.