Abstract We present an alternative proof that from large cardinals, we can force the tree property at $\kappa ^+$ and $\kappa ^{++}$ simultaneously for a singular strong limit cardinal $\kappa $ . The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $\kappa =\aleph _{\omega ^2}$ .

Abstract $ISP$ cannot hold at the first or second successor of a singular strong limit of countable cofinality; on the other hand, we force a failure of “strong ${\rm{SCH}}$ ” across a cardinal where $ITP$ holds. We also show that $ITP$ does not imply that there are stationary many internally unbounded models.

Abstract We show that from large cardinals it is consistent to have the tree property simultaneously at ${\aleph _{{\omega ^2} + 1}}$ and ${\aleph _{{\omega ^2} + 2}}$ with ${\aleph _{{\omega ^2}}}$ strong limit.

Abstract We analyze the modified extender based forcing from Assaf Sharon’s PhD thesis. We show there is a bad scale in the extension and therefore weak square fails. We also present two metatheorems which give a rough characterization of when a diagonal Prikry-type forcing forces the failure of weak square.

Abstract We show that givenωmany supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at ℵω+1·This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal andωmany supercompact cardinals above it, in Magidor–Shelah [7].

Abstract We show that givenωmany supercompact cardinals, there is a generic extension in which the tree property holds at ℵω2+ 1and the SCH fails at ℵω2.

Abstract Given a supercompact cardinalκand a regular cardinalλ<κ, we describe a type of forcing such that in the generic extension the cofinality ofκisλ, there is a very good scale atκ, a bad scale atκ, and SCH atκfails. When creating our model we have great freedom in assigning the value of 2κ, and so we can make SCH hold or fail arbitrarily badly.