On weak square, approachability, the tree property, and failures of SCH in a choiceless context

We show that the consistency of the theories ZF+¬ACω+ “GCH holds below ℵω” + “there is an injection f:ℵω+2→℘(ℵω)” + “both □ℵω∗ and APℵω fail” and ZF+¬ACω + “GCH holds below ℵω” + “there is an injection f:ℵω+2→℘(ℵω)” + “ℵω+1 satisfies the tree property” follow from the appropriate supercompactness hypotheses. These provide answers in a choiceless context to certain long‐standing open questions concerning SCH, weak square, approachability, and the tree property. There is nothing special about ℵω+2, and the injection into ℘(ℵω) can be from any ordinal λ (which means that pcf theory can fail badly in the models constructed if λ≥ℵω4).