The tree property at double successors of singular cardinals of uncountable cofinality with infinite gaps

The tree property at first and double successors of singular cardinals with an arbitrary gap

“…The present section will be devoted to introduce the main forcing construction of the paper. This forcing is a variation of the forcings appearing in [Ung13] or in [GP18], where the Supercompact Prikry/ Magidor forcing is replaced by Sinapova forcing. This modification will be, precisely, the responsible of the very good and the bad scale in our generic extension.…”

“…In particular this shows that the tree property at the double successor of a strong limit singular cardinal κ is consistent with an arbitrary failure of SCH κ . Building on [FHS18] this result was subsequently generalized in [GP18] to the setting of uncountable cofinalities as well as pushed down to the level of ℵ µ , modulo some arithmetic restrictions. 1 A related discussion to that described previously is about the existence of Aronszajn trees at first successors of strong limit singular cardinals.…”

“…In this paper we aim to combine Sinapova's arguments from [Sin16b] with those developed in [FHS18] and [GP18], in order to get a generic extension where TP(κ + ) and TP(κ ++ ) both hold, κ is a singular strong limit cardinal with cof(κ) = µ and there is an arbitrary failure of SCH κ . Further, as a consequence of results of Sinapova [Sin08], in our generic extension there will be a very good scale and a bad scale at κ.…”

“…For the proof of this result we shall make use of some ideas developed from [Sin16b] and [Sin08] for the proof of TP(κ + ) and (5), as well as some ideas from [Ung13], [FHS18] and [GP18] for the rest of the claims. The structure of the paper is as follows: In Section 2 we shall give an overview of Sinapova forcing following [Sin08] and we will prove that some suitable version of Mathias criterion for genericity holds (see e.g.…”

“…In Section 3 we will introduce our main forcing construction R and prove some basic facts about the combinatorics of its generic extensions. As the reader will notice, our forcing R is essentially the same forcing as that presented in [GP18] but here the Magidor forcing is replaced for the Sinapova forcing: this will produce the desired scales in the generic extension.…”