“School of Mathematics”
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Paper IPM / M / 16071  


Abstract:  
Assume λ is a singular limit of η supercompact cardinals, where η ≤ λ is a limit ordinal. We present two forcing methods for making
λ^{+} the successor of the limit of the first η measurable cardinals while the tree property holding at λ^{+}. The first method is then used to get,
from the same assumptions,
tree property at ℵ_{η2+1} with the failure of SCH at ℵ_{η2}. 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 MagidorShelah, Neeman and Sinapova.
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