Tree property at successor of a singular limit of measurable cardinals

DOI10.1007/S00153-017-0581-4arXiv1601.04139MaRDI QIDQ6269362

Publication date: 16 January 2016

Abstract: Assume

l a m b d a

is a singular limit of

e t a

supercompact cardinals, where

e t a l e q l a m b d a

is a limit ordinal. We present two forcing methods for making

l a m b d a^{+}

the successor of the limit of the first

e t a

measurable cardinals while the tree property holding at

l a m b d a^{+} .

The first method is then used to get, from the same assumptions, tree property at

a l e p h_{e t a^{2} + 1}

with the failure of

S C H

at

a l e p h_{e t a^{2}}

. This extends results of Neeman and Sinapova. The second method is also used to get tree property at successor of an arbitrary singular cardinal, which extends some results of Magidor-Shelah, Neeman and Sinapova.

Mathematics Subject Classification ID

Consistency and independence results (03E35) Large cardinals (03E55) Other combinatorial set theory (03E05)

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