THE INEFFABLE TREE PROPERTY and FAILURE of the SINGULAR CARDINALS HYPOTHESIS

ITP is a combinatorial principle that is a strengthening of the tree property. For an inaccessible cardinal κ, ITP at κ holds if and only if κ is supercompact. And just like the tree property, it can be forced to hold at accessible cardinals. A broad project is obtaining ITP at many cardinals simultaneously. Past a singular cardinal, this requires failure of SCH.We prove that from large cardinals, it is consistent to have failure of SCH at κ together with ITP κ+. Then we bring down the result to κ = ? ω2 .