Perfect subtree property for weakly compact cardinals

-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

“…We will now construct a tree of height that is an element of and then argue that this tree does not have a perfect subtree in . These arguments use ideas from [21] that ultimately go back to Solovay’s argument for the consistency strength of the Kurepa hypothesis (see [24, Section 4]). Our tree consists of hulls of initial segments of of size , and we will argue that we can obtain such initial segments in a -definable way with parameters in .…”

“…Following [21], we say a pair is an active node at for some if there is a good premouse N and some with , the regular cardinals in N , such that the following statements hold: x is strictly increasing and cofinal in . M is equal to the transitive collapse of and is the image of x under the transitive collapse. If is the corresponding uncollapsing map, then . In addition, we say a pair is an active node if there is some ordinal such that is an active node at .…”

“…By our first claim, it suffices to show that is a good premouse. We obtain iterability for by reflecting countable elementary substructures of into models in the tree T , as in [21], using the fact that (see [25, Lemma 3.7(ii)]). In the following, write for the sequence of models appearing in active nodes at along the branch b .…”

“…Following [21], we say a pair 𝑀, x is an active node at 𝜌 for some 𝜌 < cof (𝜅) K if there is a good premouse N and some 𝑥 ∈ cof (𝜅) 𝜅 𝑁 with ran(𝑥) ⊆ Reg 𝑁 , the regular cardinals in N, such that the following statements hold:…”

“…7 Note that the assumption that the weak compactness of a cardinal is preserved by forcing with partial orders of the form has high consistency strength (see, for example, [21] and [27]). …”